Question 1

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y). What is the velocity of the wave?

Accepted Answer

The answer of A transverse periodic wave is represented by...

Question 2

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m⁻¹ x). What is the velocity of the wave?

Accepted Answer

The wave velocity v can be found using the formula v = ω/k, where ω is the angular frequency (2,500 rad/s) and k is the wave number (15.0 m⁻¹). Thus, v = 2,500 / 15 = 167 m/s in the + x direction.

Question 3

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y). What are the wavenumber k and direction of propagation of the wave?

Accepted Answer

The wavenumber k is determined by the coefficient of y in the argument of the sine function, which is 10.0 m⁻¹. This corresponds to a wavenumber of 10 m⁻¹. The direction of propagation is given by the sign of this coefficient, which is negative, indicating that the wave is traveling in the negative y direction.

Question 4

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation z(z, t) = 1.2 cm cos(1800 rad/s t + 60 m⁻¹ z). What are the wavenumber and direction of propagation of the wave?

Accepted Answer

The wavenumber is given by the coefficient in front of z in the argument of the cosine function, which is 60 m⁻¹. Since this coefficient is negative, the wave is traveling in the − z direction.

Question 5

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m⁻¹ x). What are the wavenumber k and direction of propagation of the wave?

Accepted Answer

The general form of a transverse periodic wave is y(x,t) = A cos(kx - ωt + φ), where A is the amplitude of the wave, k is the wavenumber or wave vector, ω is the angular frequency, t is time, x is position, and φ is the phase angle.

Comparing the given equation to the general form, we can see that:
- A = 2.50 cm (amplitude)
- ω = 2,500 rad/s (angular frequency)

To find the wavenumber and direction of propagation, we need to compare the given equation to the form y(x,t) = A cos(kx - ωt). This means setting the phase angle φ to zero and rearranging the equation to isolate k:

y(x,t) = 2.50 cm cos(2,500 rad/s t − 15.0 mS-1 x)
2.50 cm = A
2,500 rad/s = ω
kx - ωt = -15.0 mS-1 x
kx + 15.0 mS-1 x = ωt
x(k + 15.0 mS-1) = ωt
k + 15.0 mS-1 = ω/t
k = (ω/t) - 15.0 mS-1
k = [2,500 rad/s] / [t]
We need to use the units that make the answer in mS-1 to match the given units. We know that the wave has a period T = 2π/ω, which is the time for one full cycle of the wave. Hence, we have:
T = 2π/ω = 2π/(2,500 rad/s) = 0.00251 s
The wavenumber is then:
k = [2,500 rad/s] / [0.00251 s] - 15.0 mS-1
k = 994.42 m-1 - 15.0 mS-1
k = 979.42 m-1 = 9.79 x 102 m-1
The direction of propagation can be determined from the sign of k. If k is positive, the wave is propagating in the + x direction, and if k is negative, the wave is propagating in the - x direction. Since k is positive in this case, the wave is traveling in the + x direction.

Therefore, the answer is B: 15.0 mS-1, traveling in the + x direction.

Question 6

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y). What is the direction of the velocity of the wave?

Accepted Answer

The direction of the velocity of a transverse wave is perpendicular to the direction of the wave's displacement (or polarization). In this case, the wave is only varying in the y-direction (as indicated by the sin term). Therefore, the velocity must be in a direction perpendicular to y, which is the x-direction. So the answer is (A) the x direction.

Question 7

A transverse periodic wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t − 20.0 m⁻¹ z). Another transverse wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t + 20.0 m⁻¹ z). What is the equation that represents the superposition of the two waves?

Accepted Answer

When two waves superimpose, the resulting displacement is given by the sum of the individual displacements. Using the identity sin(a+b) = sin(a)cos(b) + cos(a)sin(b), we can rewrite the superposition equation as y(z, t) = 2.0 cm [sin(1,200 rad/s t) cos(20.0 m^-1 z) - sin(1,200 rad/s t) cos(20.0 m^-1 z)]. Simplifying this equation gives us y(z, t) = 4.0 cm sin(1,200 rad/s t) cos(20.0 m^-1 z), which is equivalent to Choice B.

Question 8

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y). What is the direction of the vibration of the wave?

Accepted Answer

The vibration direction of a transverse wave is perpendicular to the direction of wave propagation. In the given equation, z(y, t) indicates that the wave's displacement is in the z-direction, while the wave itself propagates along the y-axis.

Question 9

A transverse periodic wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t − 20.0 m⁻¹ z). Another transverse wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t − 20.0 m⁻¹ z). What is the equation that represents the superposition of the two waves?

Accepted Answer

When two waves of the same frequency, amplitude, and wavelength superpose, their amplitudes add up, resulting in a wave with double the amplitude but the same frequency and wavelength.

Question 10

A transverse periodic wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t − 10.0 m⁻¹ x). Another transverse wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t + 10.0 m⁻¹ x). What is the equation that represents the superposition of the two waves?

Accepted Answer

The superposition of two waves with equations $y_1 = A \sin(\omega t - kx)$ and $y_2 = A \sin(\omega t + kx)$ results in $y = 2A \cos(kx) \sin(\omega t)$, where $A$ is the amplitude, $\omega$ is the angular frequency, and $k$ is the wave number. This is because of the trigonometric identity for the sum of sine functions with opposite arguments.

Question 11

A transverse periodic wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t − 10.0 m⁻¹ x). Another transverse wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t − 10.0 m⁻¹ x). What is the equation that represents the superposition of the two waves?

Accepted Answer

When two waves of the same frequency, amplitude, and wavelength superpose, their amplitudes add up. Since both given waves are identical, their superposition results in a wave with double the amplitude but the same frequency and wavelength.

Question 12

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation x(x, t) = 2.1 cm cos(2000 rad/s t − 40 m⁻¹ x). What is the direction of propagation of the wave?

Accepted Answer

The direction of propagation of a wave is given by the sign of the coefficient multiplying the x variable in the argument of the sine or cosine function. In this case, the coefficient is -40 mS-1, which is negative, indicating that the wave is propagating in the negative x direction. Therefore, the correct answer is choice B.

Question 13

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m⁻¹ x). What is the direction of the vibration of the wave?

Accepted Answer

The cosine function in the equation represents the amplitude of the wave at a given point in space (x) and time (t). The argument of the cosine function, which is (2,500 rad/s)t - 15.0 mS⁻¹x, represents the phase of the wave at that point in space and time. 
The fact that the x variable is multiplied by a negative value (-15.0 mS⁻¹) indicates that the wave is travelling in the positive x-direction. The cosine function indicates that the vibration of the wave is in the y-direction because the amplitude varies with the cosine function as opposed to the sine function, which would indicate vibration in the x-direction. Therefore, the correct answer is B, the y direction.

Question 14

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation x(x, t) = 2.1 cm cos(2000 rad/s t − 40 m⁻¹ x). What is the direction of motion of a point on the spring due to the wave?

Accepted Answer

The equation shows that the displacement of a point on the spring, x(x,t), is a function of both time, t, and position along the spring, x. The terms within the cosine function represent the angular frequency, ω = 2000 rad/s, the time, t, the wave speed, v = ω/k = 2000 rad/s / 40 m⁻¹ = 50 m/s, and the spatial position, x. The fact that the wave equation involves both cos(ωt − kx) and x suggests that the wave is moving in the x-direction, in that an observer at any fixed point on the spring will observe the wave passing by him/her in what appears to be a motion in the positive or negative x-direction. Hence, the correct answer is C.

Question 15

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y). What is the frequency of the vibration of the wave?

Accepted Answer

The answer of A transverse periodic wave is represented by...

Question 16

A transverse periodic wave is represented by the equation y(x, t) = −1.50 cm sin(1,500 rad/s t − 10.0 m⁻¹ x). Another transverse wave is represented by the equation y(x, t) = +1.50 cm sin(1,500 rad/s t + 10.0 m⁻¹ x). What is the equation that represents the superposition of the two waves?

Accepted Answer

When two waves of the same frequency, amplitude, and wavelength superpose, their amplitudes add up. Since both given waves are identical, their superposition results in a wave with double the amplitude but the same frequency and wavelength.

Question 17

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation x(x, t) = 2.10 cm cos(2000 rad/s t + 40.0 m⁻¹ x). What is the velocity of the wave?

Accepted Answer

The answer of A longitudinal wave travels on a slinky...

Question 18

The wavelength of a periodic wave is 0.500 m. If the frequency is 400 Hz, then what is the wavenumber k of the wave?

Accepted Answer

The wavenumber $k$ is given by $k = \frac{2\pi}{\lambda}$, where $\lambda$ is the wavelength. Substituting $\lambda = 0.500$ m, $k = \frac{2\pi}{0.500} = 12.57$ m$^{-1}$, which rounds to 12.6 m$^{-1}$, matching choice A.

Question 19

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m⁻¹ x). What is the frequency of the vibration of the wave?

Accepted Answer

The answer of A transverse periodic wave is represented by...

Question 20

A transverse periodic wave is represented by the equation y(x, t) = A1 sin(ωt − kx). Another transverse wave is represented by the equation y(x, t) = A2 sin(ωt + kx). What is the equation that represents the superposition of the two waves?

Accepted Answer

When two waves with the same frequency and amplitude travelling in opposite directions are superimposed, the resulting wave is given by: y(x, t) = A1 sin(ωt − kx) + A2 sin(ωt + kx)
y(x, t) = A1 sin(ωt) cos(kx) − A1 cos(ωt) sin(kx) + A2 sin(ωt) cos(kx) + A2 cos(ωt) sin(kx)
y(x, t) = (A1 + A2) sin(ωt) cos(kx) + (-A1 + A2) cos(ωt) sin(kx)
Thus, the correct option is D.

Quiz 11: Waves

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y) . What is the velocity of the wave?

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m⁻¹ x) . What is the velocity of the wave?

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y) . What are the wavenumber k and direction of propagation of the wave?

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation z(z, t) = 1.2 cm cos(1800 rad/s t + 60 m⁻¹ z) . What are the wavenumber and direction of propagation of the wave?

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m⁻¹ x) . What are the wavenumber k and direction of propagation of the wave?

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y) . What is the direction of the velocity of the wave?

A transverse periodic wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t − 20.0 m⁻¹ z) . Another transverse wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t + 20.0 m⁻¹ z) . What is the equation that represents the superposition of the two waves?

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y) . What is the direction of the vibration of the wave?

A transverse periodic wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t − 20.0 m⁻¹ z) . Another transverse wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t − 20.0 m⁻¹ z) . What is the equation that represents the superposition of the two waves?

A transverse periodic wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t − 10.0 m⁻¹ x) . Another transverse wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t + 10.0 m⁻¹ x) . What is the equation that represents the superposition of the two waves?

A transverse periodic wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t − 10.0 m⁻¹ x) . Another transverse wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t − 10.0 m⁻¹ x) . What is the equation that represents the superposition of the two waves?

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation x(x, t) = 2.1 cm cos(2000 rad/s t − 40 m⁻¹ x) . What is the direction of propagation of the wave?

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m⁻¹ x) . What is the direction of the vibration of the wave?

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation x(x, t) = 2.1 cm cos(2000 rad/s t − 40 m⁻¹ x) . What is the direction of motion of a point on the spring due to the wave?

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y) . What is the frequency of the vibration of the wave?

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation x(x, t) = 2.10 cm cos(2000 rad/s t + 40.0 m⁻¹ x) . What is the velocity of the wave?

The wavelength of a periodic wave is 0.500 m. If the frequency is 400 Hz, then what is the wavenumber k of the wave?

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m⁻¹ x) . What is the frequency of the vibration of the wave?

A transverse periodic wave is represented by the equation y(x, t) = A1 sin(ωt − kx) . Another transverse wave is represented by the equation y(x, t) = A2 sin(ωt + kx) . What is the equation that represents the superposition of the two waves?

Quiz 11: Waves

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m−1 y) . What is the velocity of the wave?

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m−1 x) . What is the velocity of the wave?

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m−1 y) . What are the wavenumber k and direction of propagation of the wave?

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation z(z, t) = 1.2 cm cos(1800 rad/s t + 60 m−1 z) . What are the wavenumber and direction of propagation of the wave?

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m−1 x) . What are the wavenumber k and direction of propagation of the wave?

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m−1 y) . What is the direction of the velocity of the wave?

A transverse periodic wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t − 20.0 m−1 z) . Another transverse wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t + 20.0 m−1 z) . What is the equation that represents the superposition of the two waves?

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m−1 y) . What is the direction of the vibration of the wave?

A transverse periodic wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t − 20.0 m−1 z) . Another transverse wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t − 20.0 m−1 z) . What is the equation that represents the superposition of the two waves?

A transverse periodic wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t − 10.0 m−1 x) . Another transverse wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t + 10.0 m−1 x) . What is the equation that represents the superposition of the two waves?

A transverse periodic wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t − 10.0 m−1 x) . Another transverse wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t − 10.0 m−1 x) . What is the equation that represents the superposition of the two waves?

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation x(x, t) = 2.1 cm cos(2000 rad/s t − 40 m−1 x) . What is the direction of propagation of the wave?

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m−1 x) . What is the direction of the vibration of the wave?

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation x(x, t) = 2.1 cm cos(2000 rad/s t − 40 m−1 x) . What is the direction of motion of a point on the spring due to the wave?

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m−1 y) . What is the frequency of the vibration of the wave?

A transverse periodic wave is represented by the equation y(x, t) = −1.50 cm sin(1,500 rad/s t − 10.0 m−1 x) . Another transverse wave is represented by the equation y(x, t) = +1.50 cm sin(1,500 rad/s t + 10.0 m−1 x) . What is the equation that represents the superposition of the two waves?

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation x(x, t) = 2.10 cm cos(2000 rad/s t + 40.0 m−1 x) . What is the velocity of the wave?

The wavelength of a periodic wave is 0.500 m. If the frequency is 400 Hz, then what is the wavenumber k of the wave?

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m−1 x) . What is the frequency of the vibration of the wave?

A transverse periodic wave is represented by the equation y(x, t) = A1 sin(ωt − kx) . Another transverse wave is represented by the equation y(x, t) = A2 sin(ωt + kx) . What is the equation that represents the superposition of the two waves?

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y) . What is the velocity of the wave?

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m⁻¹ x) . What is the velocity of the wave?

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y) . What are the wavenumber k and direction of propagation of the wave?

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation z(z, t) = 1.2 cm cos(1800 rad/s t + 60 m⁻¹ z) . What are the wavenumber and direction of propagation of the wave?

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m⁻¹ x) . What are the wavenumber k and direction of propagation of the wave?

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y) . What is the direction of the velocity of the wave?

A transverse periodic wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t − 20.0 m⁻¹ z) . Another transverse wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t + 20.0 m⁻¹ z) . What is the equation that represents the superposition of the two waves?

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y) . What is the direction of the vibration of the wave?

A transverse periodic wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t − 20.0 m⁻¹ z) . Another transverse wave is represented by the equation y(z, t) = 2.0 cm sin(1,200 rad/s t − 20.0 m⁻¹ z) . What is the equation that represents the superposition of the two waves?

A transverse periodic wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t − 10.0 m⁻¹ x) . Another transverse wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t + 10.0 m⁻¹ x) . What is the equation that represents the superposition of the two waves?

A transverse periodic wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t − 10.0 m⁻¹ x) . Another transverse wave is represented by the equation y(x, t) = 1.50 cm sin(1,500 rad/s t − 10.0 m⁻¹ x) . What is the equation that represents the superposition of the two waves?

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation x(x, t) = 2.1 cm cos(2000 rad/s t − 40 m⁻¹ x) . What is the direction of propagation of the wave?

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m⁻¹ x) . What is the direction of the vibration of the wave?

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation x(x, t) = 2.1 cm cos(2000 rad/s t − 40 m⁻¹ x) . What is the direction of motion of a point on the spring due to the wave?

A transverse periodic wave is represented by the equation z(y, t) = 1.50 cm sin(1,250 rad/s t + 10.0 m⁻¹ y) . What is the frequency of the vibration of the wave?

A longitudinal wave travels on a slinky or any long spring. The wave is represented by the equation x(x, t) = 2.10 cm cos(2000 rad/s t + 40.0 m⁻¹ x) . What is the velocity of the wave?

A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m⁻¹ x) . What is the frequency of the vibration of the wave?