Question

To understand the relationships among the parameters that characterize a wave. It is of fundamental importance in many areas of physics to be able to deal with waves. This problem will lead you to understand the relationship of variables related to wave propagation: frequency, wavelength, velocity of propagation, and related variables. Note that these are kinematic variables that relate to the wave's propagation and do not depend on its amplitude.

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A. Traveling waves propagate with a fixed speed usually denoted as v (but sometimes c). The waves are called __________ if their waveform repeats every time interval T.

-transverse

-longitudinal

-periodic

-sinusoidal

B. Solve this equation to find an expression for the wavelength ?.

C. If the velocity of the wave remains constant, then as the frequency of the wave is increased, the wavelength __________.

-decreases

-increases

-stays constant

D. The difference between the frequency f and the frequency ? is that f is measured in cycles per second or hertz (abbreviated Hz) whereas the units for ? are __________ per second.

E. Find an expression for the period T of a wave in terms of other kinematic variables.

F. What is the relationship between ? and f?

Answer 1

Concepts and reason

The concepts used in this problem are travelling waves, frequency, wavelength, and velocity of waves.

First, find type of the wave motion by analyzing the wave characteristics. Then, find the wavelength of the wave by using the expression of the expression for the frequency. Then, find the relation between frequency and wavelength to find the change in wavelength. After that, find the units of the angular frequency by using its expression. Later, find the time-period of the wave motion using the expression of frequency. Finally, find the expression of the angular frequency using the expression of frequency.

Fundamentals

The expression for the frequency of the wavelength is as follows:

$f = \frac{v}{\lambda }$

Here, v is the speed of wave and $\lambda$ is the wavelength of the wave.

The time-period of the wave propagation is as follows:

$T = \frac{1}{f}$

Here, f is the frequency of the wave motion.

(A)

A travelling wave moving with a phase velocity of v completes a full cycle in time T. this means that the wave form completes itself in the interval of T time. The next wave form will also complete itself in next T time. This will go on forever as the wave is moving with a constant velocity v.

The motion of the wave is repeating itself in time T. This type of motion is called as periodic motion. The period of this motion is T. the waves are called as periodic waves.

(B)

The expression for the frequency of the wavelength is as follows:

$f = \frac{v}{\lambda }$

Rearrange the above expression for wavelength.

$\lambda = \frac{v}{f}$

(C)

The wavelength of a wave is given as follows:

$\lambda = \frac{v}{f}$

The velocity of the wave remains constant and is equal to v.

The initial wavelength is given as follows:

${\lambda _1} = \frac{v}{{{f_1}}}$

The final wavelength is given as follows:

${\lambda _2} = \frac{v}{{{f_2}}}$

Divide equation ${\lambda _2} = \frac{v}{{{f_2}}}$ by ${\lambda _1} = \frac{v}{{{f_1}}}$ and find the expression of ${\lambda _2}$ .

$\begin{array}{c}\\\frac{{{\lambda _2}}}{{{\lambda _1}}} = \left( {\frac{v}{{{f_2}}}} \right)\left( {\frac{{{f_1}}}{v}} \right)\\\\{\lambda _2} = \left( {\frac{{{f_1}}}{{{f_2}}}} \right){\lambda _1}\\\end{array}$

Thus, the wavelength decreases as the frequency increases.

(D)

The expression of the angular frequency is as follows:

$\omega = \left( {2\pi {\rm{ rad}}} \right)f$

Here, f is the frequency of the wave.

Mention the units of frequency in the above expression to find the units of $\omega$ . Substitute $f{\rm{ }}{{\rm{s}}^{ - 1}}$ for f in the above expression.

$\begin{array}{c}\\\omega = \left( {2\pi {\rm{ rad}}} \right)\left( {f{\rm{ }}{{\rm{s}}^{ - 1}}} \right)\\\\ = \left( {2\pi f} \right){\rm{ rad/s}}\\\end{array}$

Thus, the units of $\omega$ is radians per second.

(E)

The time period of propagation of wave is given by the following expression:

$T = \frac{1}{f}$

Here, f is the frequency of the wave.

(F)

The wave completes its one cycle in time T. the angular frequency is $\omega$ . The total angle completed by the wave in one cycle the product of angular frequency and time period. The angle covered by wave in one cycle can be written as follows:

$\omega T = 2\pi$

Rearrange the above expression for $\omega$ .

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

Substitute $\frac{1}{f}$ for T in the above expression.

$\omega = 2\pi f$

Ans: Part A

The waves are called periodic waves.

Part B

The wavelength is $\lambda = \frac{v}{f}$ .

Part C

The wavelength decreases.

Part D

The units of $\omega$ is radians per second.

Part E

The time-period is $\frac{1}{f}$ .

Part F

The angular frequency is $2\pi f$ .