Question

A normal mode of a closed system is an oscillation of the system in which all parts oscillate at a single frequency. In general there are an infinite number of such modes, each one with a distinctive frequency fi and associated pattern of oscillation.

Consider an example of a system with normal modes: a string of length L held fixed at both ends, located at x=0 and x=L. Assume that waves on this string propagate with speed v. The string extends in the x direction, and the waves are transverse with displacement along the y direction.

In this problem, you will investigate the shape of the normal modes and then their frequency.

The normal modes of this system are products of trigonometric functions. (For linear systems, the time dependance of a normal mode is always sinusoidal, but the spatial dependence need not be.) Specifically, for this system a normal mode is described by

yi(x,t)=Ai sin(2π*x/λi)sin(2πfi*t)

A)The string described in the problem introduction is oscillating in one of its normal modes. Which of the following statements about the wave in the string is correct?

The wave is traveling in the +x direction.
a) The wave is traveling in the -x direction.
b) The wave will satisfy the given boundary conditions for any arbitrary wavelength lambda_i.
c) The wavelength lambda_i can have only certain specific values if the boundary conditions are to be satisfied.
d) The wave does not satisfy the boundary condition y_i(0;t)=0.
B)Which of the following statements are true?

a)The system can resonate at only certain resonance frequencies f_i and the wavelength lambda_i must be such that y_i(0;t) = y_i(L;t) = 0.
b) A_i must be chosen so that the wave fits exactly on the string.
c) Any one of A_i or lambda_i or f_i can be chosen to make the solution a normal mode.

C) Find the three longest wavelengths (call them lambda_1, lambda_2, and lambda_3) that "fit" on the string, that is, those that satisfy the boundary conditions at x=0 and x=L. These longest wavelengths have the lowest frequencies.

D) The frequency of each normal mode depends on the spatial part of the wave function, which is characterized by its wavelength lambda_i.

Find the frequency f_i of the ith normal mode.

Answer 1

Answer:

Explanation:

(A)

The string has set of normal modes and the string is oscillating in one of its modes.

The resonant frequencies of a physical object depend on its material, structure and boundary conditions.

The free motion described by the normal modes take place at the fixed frequencies and these frequencies is called resonant frequencies.

Given below are the incorrect options about the wave in the string.

• The wave is travelling in the +x direction

• The wave is travelling in the -x direction

• The wave will satisfy the given boundary conditions for any arbitrary wavelength $\lambda_i$

• The wave does not satisfy the boundary conditions $y_i(0;t)=0$

Here, the string of length L held fixed at both ends, located at x=0 and x=L

The key constraint with normal modes is that there are two spatial boundary conditions,$y(0,1)=0$

and $y(L,t)=0$

.The spring is fixed at its two ends.

The correct options about the wave in the string is

• The wavelength $\lambda_i$  can have only certain specific values if the boundary conditions are to be satisfied.

(B)

The key factors producing the normal mode is that there are two spatial boundary conditions, $y_i(0;t)=0$ and $y_i(L;t)=0$, that are satisfied only for particular value of $\lambda_i$  .

Given below are the incorrect options about the wave in the string.

•  $A_i$ must be chosen so that the wave fits exactly o the string.

• Any one of  $A_i$ or $\lambda_i$  or $f_i$  can be chosen to make the solution a normal mode.

Hence, the correct option is that the system can resonate at only certain resonance frequencies $f_i$ and the wavelength $\lambda_i$  must be such that $y_i(0;t) = y_i(L;t)=0$

(C)

Expression for the wavelength of the various normal modes for a string is,

$\lambda_n=\frac{2L}{n}$ (1)

When $n=1$ , this is the longest wavelength mode.

Substitute 1 for n in equation (1).

$\lambda_n=\frac{2L}{1}\\\\2L$

When $n=2$ , this is the second longest wavelength mode.

Substitute 2 for n in equation (1).

$\lambda_n=\frac{2L}{2}\\\\L$

When $n=3$, this is the third longest wavelength mode.

Substitute 3 for n in equation (1).

$\lambda_n=\frac{2L}{3}$

Therefore, the three longest wavelengths are $2L$,$L$ and $\frac{2L}{3}$.

(D)

Expression for the frequency of the various normal modes for a string is,

$f_n=\frac{v}{\lambda_n}$

For the case of frequency of the $i^{th}$ normal mode the above equation becomes.

$f_i=\frac{v}{\lambda_i}$

Here, $f_i$ is the frequency of the $i^{th}$ normal mode, v is wave speed, and $\lambda_i$ is the wavelength of $i^{th}$ normal mode.

Therefore, the frequency of $i^{th}$ normal mode is  $f_i=\frac{v}{\lambda_i}$

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