Question:

A transverse wave propagating along a stretched string has the following characteristics:

• Amplitude $A$ = 0.2 m
• Wavelength $\lambda$ = 0.6 m
• Frequency $f$ = 80 Hz
• Wave speed $v$ = 48 m/s

A) Calculate the angular frequency $\omega$ of the wave.

B) Determine the period $T$ of the wave.

C) Find the displacement $y$ of a point on the string located 0.4 m away from the wave source at time $t = 2$ s.

Answer:

A)

The angular frequency, $\omega$, can be calculated using the formula:

Given that the frequency $f$ is 80 Hz, we can substitute the value in the formula:

Therefore, the angular frequency $\omega$ of the wave is $160\pi \ \text{rad/s}$.

B)

The period $T$ of a wave can be calculated using the formula:

Given that the frequency $f$ is 80 Hz, we can substitute the value in the formula:

Therefore, the period $T$ of the wave is $0.0125 \ \text{s}$.

C)

To find the displacement $y$ of a point on the string located 0.4 m away from the wave source at time $t = 2$ s, we can use the equation:

Where:

• $A$ is the amplitude of the wave
• $k = \frac{2\pi}{\lambda}$ is the wave number
• $x$ is the position of the point on the string
• $\omega$ is the angular frequency of the wave
• $t$ is the time
• $\phi$ is the phase constant (which we can assume to be zero for simplicity in this case)

Given that the amplitude $A$ is 0.2 m, wavelength $\lambda$ is 0.6 m, angular frequency $\omega$ is $160\pi \ \text{rad/s}$, and time $t$ is 2 s, we can substitute these values in the equation:

Simplifying,

Therefore, the displacement $y$ of a point on the string located 0.4 m away from the wave source at time $t = 2$ s is approximately 0.173 m.