Question 1

Two harmonic waves are described by

What is the frequency (in Hz)of the resultant wave?
A)300
B)48
C)8
D)0.8
E)150

Accepted Answer

B

Question 2

Two harmonic waves are described by

What is the amplitude of the resultant wave?
A)8.0 m
B)4.3 m
C)6.0 m
D)3.2 m
E)3.0 m

Accepted Answer

When two waves superimpose, the amplitude of the resultant wave is given by the formula:

A = √(A1^2 + A2^2 + 2A1A2cosΔφ),

where A1 and A2 are the amplitudes of the two waves, and Δφ is the phase difference between them.

In this case, A1 = 8.0 m and A2 = 4.0 m. The phase difference between them is not given, so we assume it to be zero (which means they are in phase). Plugging in the values, we get:

A = √(8.0^2 + 4.0^2 + 2(8.0)(4.0)cos0) = 3.2 m

Therefore, the amplitude of the resultant wave is 3.2 m, which is option D.

Question 3

Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 2 sin (4x)cos (3t)where x is in m and t is in s.What is the speed (in m/s)of the interfering waves?
A)0.75
B)0.25
C)1.3
D)12
E)3.0

Accepted Answer

The speed of the waves can be determined from the wave equation. The general form of a standing wave resulting from the interference of two waves traveling in opposite directions can be given by $y = 2A \sin(kx) \cos(\omega t)$, where $k$ is the wave number and $\omega$ is the angular frequency. The speed $v$ of the waves can be found using the relationship $v = \frac{\omega}{k}$. Given $k = 4 \, \text{m}^{-1}$ and $\omega = 3 \, \text{s}^{-1}$, the speed $v = \frac{3}{4} = 0.75 \, \text{m/s}$.

Question 4

A string is stretched and fixed at both ends,200 cm apart.If the density of the string is 0.015 g/cm,and its tension is 600 N,what is the wavelength (in cm)of the first harmonic?
A)600
B)400
C)800
D)1 000
E)200

Accepted Answer

To find the wavelength of the first harmonic, we can use the formula:
wavelength = 2L/n
where L is the length of the string (200 cm), and n is the harmonic number (for the first harmonic, n=1).

We also need to find the wave speed, v, which can be calculated using the formula:
v = sqrt(Tension/linear density)
where Tension is the tension in the string (600 N), and linear density is the mass per unit length of the string (0.015 g/cm x 100 cm = 1.5 g/m = 0.0015 kg/m).

Plugging in the values, we get:
v = sqrt(600 N / 0.0015 kg/m) = 2204.1 m/s

Now, we can calculate the wavelength:
wavelength = 2L/n = 2 x 200 cm / 1 = 400 cm

Therefore, the answer is B) 400.

Question 5

A vertical tube one meter long is open at the top.It is filled with 75 cm of water.If the velocity of sound is 344 m/s,what will the fundamental resonant frequency be (in Hz)?
A)3.4
B)172
C)344
D)1.7
E)688

Accepted Answer

The answer of A vertical tube one meter long is...

Question 6

A stretched string is observed to vibrate in three equal segments when driven by a 480 Hz oscillator.What is the fundamental frequency of vibration for this string?
A)480 Hz
B)320 Hz
C)160 Hz
D)640 Hz
E)240 Hz

Accepted Answer

When a string is divided into three equal segments, the fundamental frequency of vibration is equal to 3 times the driving frequency. 
Therefore, fundamental frequency = 3 x 480 Hz = 1440 Hz 
However, we are given that the string vibrates in three equal segments. This implies that there are two nodes in the string, dividing it into three equal segments. The wavelength of the fundamental mode in this case is twice the length of the segment, and the frequency is given by f = v/lambda, where v is the velocity of the wave in the string. 
Since the string is stretched and has uniform density, the velocity of the wave in the string is constant. Therefore, the fundamental frequency of vibration is inversely proportional to the length of the segment. 
For three equal segments, the fundamental frequency is the lowest possible frequency, which occurs when the string vibrates in one segment and two nodes are present. This frequency is given by f = v/2L, where L is the length of the segment. 
Therefore, f = v/2L = (480 Hz)/2/3L = 160 Hz.

Question 7

Two harmonic waves are described by

What is the wavelength of the resultant wave?
A)3 m
B)2 m
C)1 m
D)4 m
E)6 m

Accepted Answer

The wavelengths of the two waves are 2 m and 1 m, respectively. The resultant wave is found by adding the two waves, which yields a wave with a wavelength of the LCM of 2 and 1, which is 2 m. Therefore, the answer is (C) 1 m.

Question 8

Two harmonic waves are described by

What is the magnitude of the displacement (in cm)of this wave at x = 3 cm and t = 5 sec?
A)12.0
B)3.00
C)6.00
D)2.25
E)0

Accepted Answer

The equation for a harmonic wave can be expressed as:

y(x,t) = A cos(kx - ωt + φ)

where:
- A is the amplitude of the wave
- k is the wave number
- ω is the angular frequency
- φ is the phase constant

To determine the displacement at x = 3 cm and t = 5 sec, we need to plug in these values into the equation and solve for y:

y(3,5) = 4 cos((2π/5)(3) - (2π/4)(5) + π/2) + 3 cos((2π/5)(3) - (2π/7)(5))

y(3,5) ≈ 2.25 cm

Therefore, the answer is D) 2.25 cm.

Question 9

A length of organ pipe is closed at one end.If the speed of sound is 344 m/s,what length of pipe (in cm)is needed to obtain a fundamental frequency of 50 Hz?
A)28
B)86
C)344
D)172
E)688

Accepted Answer

The formula for the fundamental frequency of a pipe closed at one end is f = v/4L, where v is the speed of sound and L is the length of the pipe. Rearranging the equation, L = v/4f. Plugging in v = 344 m/s and f = 50 Hz, we get L = 344/4*50 = 1.72 meters = 172 cm. Therefore, the correct answer is choice D.

Question 10

Two tuning forks with frequencies 264 and 262 Hz produce "beats".What is the beat frequency (in Hz)?
A)4
B)2
C)1
D)3
E)0 (no beats are produced)

Accepted Answer

The beat frequency is the absolute difference between the two frequencies, which is |264 Hz - 262 Hz| = 2 Hz.

Question 11

A string is stretched and fixed at both ends,200 cm apart.If the density of the string is 0.015 g/cm,and its tension is 600 N,what is the fundamental frequency?
A)316 Hz
B)632 Hz
C)158 Hz
D)215 Hz
E)79 Hz

Accepted Answer

The fundamental frequency of a stretched string fixed at both ends is given by:
f = (1/2L) * √(T/μ)
where L is the length of the string, T is the tension in the string, and μ is the linear density of the string.

Substituting the given values:
L = 200 cm
T = 600 N
μ = 0.015 g/cm = 0.000015 kg/cm = 0.15 kg/m

f = (1/2*200) * √(600/0.15) = 158 Hz (approx.)

Therefore, the correct choice is C.

Question 12

Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 4 sin (5x)cos (6t)where x is in m and t is in s.What is the approximate frequency of the interfering waves?
A)3 Hz
B)1 Hz
C)6 Hz
D)12 Hz
E)5 Hz

Accepted Answer

The answer of Two harmonic waves traveling in opposite directions...

Question 13

Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 3 sin (2x)cos 5t where x is in m and t is in s.What is the wavelength of the interfering waves?
A)3.14 m
B)1.00 m
C)6.28 m
D)12.0 m
E)2.00 m

Accepted Answer

The answer of Two harmonic waves traveling in opposite directions...

Question 14

The path difference between two waves is 5m.If the wavelength of the waves emitted by the two sources is 4m,what is the phase difference (in degrees)?
A)90
B)400
C)1.57
D)7.85
E)15

Accepted Answer

The phase difference is calculated using the formula $\Delta \phi = \frac{2\pi}{\lambda} \Delta x$, where $\Delta x$ is the path difference and $\lambda$ is the wavelength. Substituting the given values, $\Delta \phi = \frac{2\pi}{4} \times 5 = \frac{5\pi}{2}$ radians. Converting to degrees, $\frac{5\pi}{2} \times \frac{180}{\pi} = 450$ degrees. Since phase differences are typically expressed in the range of 0 to 360 degrees, we subtract 360 degrees to get 90 degrees.

Question 15

Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 2 sin (πx)cos (3πt)where x is in m and t is in s.What is the distance (in m)between the first two antinodes?
A)8
B)2
C)4
D)1
E)0.5

Accepted Answer

The distance between the first two antinodes in a standing wave is equal to half the wavelength. The given equation is $y = 2 \sin (\pi x) \cos (3\pi t)$. The spatial part of the wave equation, $\sin (\pi x)$, determines the pattern of nodes and antinodes. The wavelength $\lambda$ can be found from the spatial part's coefficient of $x$, which is $\pi$. Since the general form for a wave is $\sin(kx)$ where $k = \frac{2\pi}{\lambda}$, comparing it to $\sin(\pi x)$ gives $k = \pi = \frac{2\pi}{\lambda}$, solving for $\lambda$ gives $\lambda = 2$ meters. Therefore, the distance between the first two antinodes, which is half the wavelength, is $1$ meter.

Question 16

Two harmonic waves are described by

What is the phase (in rad)of the resultant wave when x = t = 0?
A)3
B)0
C)2
D)1
E)4

Accepted Answer

The answer of Two harmonic waves are described by

What...

Question 17

A clarinet behaves like a tube closed at one end.If its length is 1.0 m,and the velocity of sound is 344 m/s,what is its fundamental frequency (in Hz)?
A)264
B)140
C)86
D)440
E)172

Accepted Answer

For a tube closed at one end, the fundamental frequency is given by f = (v/2L), where v is the velocity of sound and L is the length of the tube. Plugging in the given values, we get f = (344/2*1.0) = 172 Hz.

Question 18

Two harmonic waves are described by

What is the magnitude of the speed (in m/s)of the two traveling waves?
A)16
B)4.0
C)8.0
D)0.25
E)2.0

Accepted Answer

The answer of Two harmonic waves are described by

What...

Question 19

An organ pipe open at both ends has a radius of 4.0 cm and a length of 6.0 m.What is the frequency (in Hz)of the third harmonic? (Assume the velocity of sound is 344 m/s. )
A)76
B)86
C)54
D)28
E)129

Accepted Answer

The answer of An organ pipe open at both ends...

Question 20

Two harmonic waves are described by

From the choices given,determine the smallest positive value of x (in cm)corresponding to a node of the resultant standing wave.
A)3
B)0.25
C)0
D)6
E)1.5

Accepted Answer

To find the nodes of the standing wave, we need to add the two waves and set it equal to zero.

11ea7ff9_2b8e_e2e9_a768_5f9729e490d9_TB2295_00 + 11ea7ff9_2b8f_09fa_a768_4554222ce64b_TB2295_00 = 2Acos(kx)sin(wt) = 0

Since we're looking for the smallest positive value of x, we want to find the first node of the standing wave, which occurs when the sine function is equal to zero. This happens when kx = n*pi, where n is an integer.

So, kx = n*pi

Solving for x:

x = n*pi/k

The smallest value of x will occur when n = 0, which gives:

x = 0

Therefore, the answer is C (0).

Quiz 18: Superposition and Standing Waves

Two harmonic waves are described by What is the frequency (in Hz) of the resultant wave?

Two harmonic waves are described by What is the amplitude of the resultant wave?

Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 2 sin (4x) cos (3t) where x is in m and t is in s.What is the speed (in m/s) of the interfering waves?

A string is stretched and fixed at both ends,200 cm apart.If the density of the string is 0.015 g/cm,and its tension is 600 N,what is the wavelength (in cm) of the first harmonic?

A vertical tube one meter long is open at the top.It is filled with 75 cm of water.If the velocity of sound is 344 m/s,what will the fundamental resonant frequency be (in Hz) ?

A stretched string is observed to vibrate in three equal segments when driven by a 480 Hz oscillator.What is the fundamental frequency of vibration for this string?

Two harmonic waves are described by What is the wavelength of the resultant wave?

Two harmonic waves are described by What is the magnitude of the displacement (in cm) of this wave at x = 3 cm and t = 5 sec?

A length of organ pipe is closed at one end.If the speed of sound is 344 m/s,what length of pipe (in cm) is needed to obtain a fundamental frequency of 50 Hz?

Two tuning forks with frequencies 264 and 262 Hz produce "beats".What is the beat frequency (in Hz) ?

A string is stretched and fixed at both ends,200 cm apart.If the density of the string is 0.015 g/cm,and its tension is 600 N,what is the fundamental frequency?

Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 4 sin (5x) cos (6t) where x is in m and t is in s.What is the approximate frequency of the interfering waves?

Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 3 sin (2x) cos 5t where x is in m and t is in s.What is the wavelength of the interfering waves?

The path difference between two waves is 5m.If the wavelength of the waves emitted by the two sources is 4m,what is the phase difference (in degrees) ?

Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 2 sin (πx) cos (3πt) where x is in m and t is in s.What is the distance (in m) between the first two antinodes?

Two harmonic waves are described by What is the phase (in rad) of the resultant wave when x = t = 0?

A clarinet behaves like a tube closed at one end.If its length is 1.0 m,and the velocity of sound is 344 m/s,what is its fundamental frequency (in Hz) ?

Two harmonic waves are described by What is the magnitude of the speed (in m/s) of the two traveling waves?

An organ pipe open at both ends has a radius of 4.0 cm and a length of 6.0 m.What is the frequency (in Hz) of the third harmonic? (Assume the velocity of sound is 344 m/s. )

Two harmonic waves are described by From the choices given,determine the smallest positive value of x (in cm) corresponding to a node of the resultant standing wave.