Sinusoidal waves travel on five identical strings.Four of the strings have the same tension, but the fifth has a different tension.Use the mathematical forms of the waves, gives below, to identify the string with the different tension.In the expressions given below x and y are in centimeters and t is in seconds.
A)y(x,t)= (2 cm)sin (2x - 4t)
B)y(x,t)= (2 cm)sin (4x - 10t)
C)y(x,t)= (2 cm)sin (6x - 12t)
D)y(x,t)= (2 cm)sin (8x - 16t)
E)y(x,t)= (2 cm)sin (10x - 20t)

Question

Quizplus · Accepted Answer

The wave speed $v$ in a string is given by $v = \sqrt{\frac{T}{\mu}}$, where $T$ is the tension in the string and $\mu$ is the mass per unit length of the string. The phase velocity of a wave is given by $v = \frac{\omega}{k}$, where $\omega$ is the angular frequency and $k$ is the wave number. In the given equations, the term in the sine function can be written as $kx - \omega t$, where $k$ is the coefficient of $x$ and $\omega$ is the coefficient of $t$. For strings A, C, D, and E, the ratio $\frac{\omega}{k}$ is constant (2 for A, 2 for C, 2 for D, and 2 for E), indicating the same wave speed and thus the same tension. However, for string B, this ratio is $\frac{10}{4} = 2.5$, which is different, indicating a different tension in the string.

Sinusoidal Waves Travel on Five Identical Strings