The equation of a plane progressive wave is given by

y =5cosπ (200t− x/ 150)

where x and y are in cm and t is in seconds. Find the wave velocity.

The correct answer is 300m/s.

To find the wave velocity, we compare the given wave equation with the standard equation of a progressive wave.

The given equation of the plane progressive wave is:

$y=5cos\pi \left(200t-\frac{x}{150}\right)$

By distributing the term $\pi$ inside the parentheses, we can rewrite the equation as:

$y=5cos\left(200\pi t-\frac{\pi x}{150}\right)$

The standard equation of a plane progressive wave traveling in the positive x-direction is:

$y=Acos(\omega t-kx)$

where:
- $A$ is the amplitude of the wave,
- $\omega$ is the angular frequency, and
- $k$ is the wave number (propagation constant).

By comparing the terms of both equations, we get:

$\omega =200\pi \text{rad/s}$

and

$k=\frac{\pi }{150}\text{rad/cm}$

The wave velocity ($v$) is related to the angular frequency ($\omega$) and the wave number ($k$) by the formula:

$v=\frac{\omega }{k}$

Substituting the values of $\omega$ and $k$ into the formula:

$v=\frac{200\pi }{\pi /150}$

Simplifying the fraction:

$v=200\times 150\text{cm/s}$

$v=30000\text{cm/s}$

To convert the wave velocity from centimeters per second (cm/s) to meters per second (m/s), we divide by 100:

$v=\frac{30000}{100}\text{m/s}=300\text{m/s}$

Therefore, the wave velocity is 300m/s.