Question

When a wave crosses a boundary between thin and thick rope, its wavelength and velocity change, but its frequency does not. Why is the frequency constant?

Answer 1

Hint: Students should refer to the definition of frequency, wavelength and velocity. We have to differentiate between the two strings based on their diameters and find the velocity and frequency based on this.

Complete step by step answer:
The tension for the two strings is the same. The formula for the propagation of sound waves on a stretched string is
$\upsilon = \sqrt {\dfrac{T}{m}} = \sqrt {\dfrac{T}{{A\rho }}} $
where $A$ is the area of the cross section of a string and $\rho $ is mass per unit volume of the string.
$\rho = \dfrac{M}{{Al}} = \dfrac{m}{A}$
By this relation it says that the more the cross-sectional area the less will be the wave velocity.When it crosses from one string to another the velocity of the wave changes as the area of the material has changed.As per the formula,

The frequency of stationary waves on a string of length $L$:
\[v = \dfrac{1}{L} = \sqrt {\dfrac{T}{m}} \]
\[\Rightarrow v = \dfrac{1}{L} = \sqrt {\dfrac{T}{\mu }} \]
Here, $L$ is the length of the entire system and $\mu $ is the mass per unit length for the entire system.
This frequency is the same for the entire system as it will be produced when the two ends of the composite string will be tied to form stationary waves. As the frequency is same for both the material and the velocity is different.
$\upsilon = v\lambda $
As the velocity changes the wavelength also gets changed as it is directly proportional to each other. So, we can say that the frequency is constant when it crosses from one material to the other.

Note: Frequency is defined as the number of oscillations or occurrences per unit time. The unit of frequency is Hertz. Frequency is also defined as the reciprocal of time period. From the above calculation you can see that by increasing the length and diameter of the cross-section of wire the frequency decreases.