Question

The displacement of a wave traveling in the negative y-direction is $D(y,t)=(5.2cm)\sin (5.5y+72t)$, where y is in m and t is in s.

What are the (a) frequency, (b) wavelength, and (c) speed of this wave?

Short answer

Part (a) The frequency of the wave is 11.459 Hz.

Part (b) The wavelength is 1.142 m.

Part (c) The wave speed is 13.01 m/s.

Step-by-step solution

Part (a) Step 1. Given information

The wavenumber of the wave =k =5.5 rad/m

The angular frequency of the wave =$\omega$=72 rad/s

Part (a) Step 2. Calculating the frequency:

The expression for the wave function of a sinusoidal wave propagating along Y-axis is given by

The angular frequency is defined as$2\pi$times the frequency of the wave.

The formula to calculate the frequency is given by

$f=\frac{\omega }{2\pi }$

Substitute 72 rad/s for $\omega$to calculate the frequency.

$$\begin{gathered} f=\frac{72rad/s}{2\times \pi } \\ =11.459Hz \end{gathered}$$

Therefore, the frequency of the wave is 11.459 Hz.

Part (b) Step 1. Calculating the wavelength:

The distance between two consecutive crests or troughs is called the wavelength.

The formula to calculate the wavelength in terms of wave number is given by

$\lambda =\frac{2\pi }{k}$

Substitute 5.5 rad/m for k into the above formula to calculate the wavelength.

$$\begin{gathered} \lambda =\frac{2\pi }{5.5rad/m} \\ =1.142m \end{gathered}$$

Therefore, the wavelength is 1.142 m.

Part (c) Step 1. Calculating the wave speed of the wave:

For a periodic wave, the wave speed is the product of frequency and wavelength of the wave and it is written as $v=f\lambda$

Substitute 11.459 Hz for f and 1.142 m for $\lambda$to calculate the speed of the wave.

$$\begin{gathered} v=11.459Hz\times 1.142m \\ =13.01m/s \end{gathered}$$

Therefore, the wave speed is 13.01 m/s.