Standing Waves

We consider a stretched string of length $L$ as shown in Figure 15.8. The string is plucked and left to vibrate. In this case we have a transverse wave where $\xi(x,t)$ the displacement of the string is perpendicular to the direction of the string which is along the $x$ axis.

Figure 15.8: A string fixed at the ends

As we have seen in the earlier section that the evolution of $\xi(x,t)$ is governed by the wave equation,

$$\frac{\partial^2 \xi}{\partial x^2} - \frac{1}{c_s^2 } \ \frac{\partial^2 \xi}{\partial t^2} = 0,$$ (15.45)

where $c_s=\sqrt{T/\mu}$. Here $T$ is the tension in the string and $\mu$ is the mass per unit length of the string. The two ends of the string are fixed. This imposes the boundary conditions $\xi(0,t)=0$ and $\xi(L,t)=0$. We could proceed by taking the general form of the solution and imposing the boundary conditions. Instead of doing this we proceed to introduce a different method of solving the wave equation. We take a trial solution of the form,

$$\xi(x,t)= X(t) T(t),$$ (15.46)

ie. $\xi(x,t)$ is the product of two functions, the function $X(x)$ depends only on $x$ and the function $T(t)$ depends only on $t$. This is referred to as the separation of variables. The wave equation now reads,

$$T \frac{d^2 X}{d x^2}-\frac{X}{c_s^2} \frac{d^2 T}{d t^2}=0.$$ (15.47)

We divide this equation by $XT$, which gives,

$$\frac{1}{X} \frac{d^2 X}{d x^2}= \frac{1}{T c_s^2} \frac{d^2 T}{d t^2}.$$ (15.48)

The left hand side of this equation is a function of $x$ alone whereas the right hand side is a function of $t$ alone. This implies that each of these two should be separately equal to a constant ie.

$$\frac{1}{X} \frac{d^2 X}{d x^2}=\alpha,$$ (15.49)

and

$$\frac{1}{T c_s^2} \frac{d^2 T}{d t^2}=\alpha \,.$$ (15.50)

Let us first consider the solution to $X(x)$. These are of the form,

$$X(x)=B_1 e^{\sqrt{\alpha} x} + B_2 e^{-\sqrt{\alpha} x}.$$ (15.51)

In the situation where $\alpha>0$, it is not possible to simultaneously satisfy the two boundary condition that $\xi(0,t)=0$ and $\xi(L,t)=0$. We therefore consider $\alpha<0$ and write it as,

$$\alpha=-k^2,$$ (15.52)

and the equation governing $X(x)$ is

$$\frac{d^2 X}{d x^2}=- k^2 X \,.$$ (15.53)

This is the familiar differential equation of a Simple Harmonic Oscillator whose solution is,

$$X(x)=A \cos(k x + \psi) \,.$$ (15.54)

The boundary condition $X(0)=0$ implies that $\psi=\pm \pi/2$ whereby

$$X(x)=A \sin(k x ) \,.$$ (15.55)

The boundary condition $X(L)=0$ is satisfied only if,

$$k=\frac{N \pi}{L}, \, \hspace{1cm} \, (N=1,2,3,...)\,.$$ (15.56)

We see that there are a large number of possible solutions, one corresponding to each value of the integer $N=1,2,3,...$. Let us next consider the time dependence which is governed by,

$$\frac{d^2 T }{d t^2} = - c_s^2 k^2 T,$$ (15.57)

which has a solution,

$$T(t)=B \cos(\omega t + \phi),$$ (15.58)

where $\omega=c_s k$. Combining $X(x)$ and $T(t)$ we obtain the solution,

$$\xi(x,t)=A_N \sin(k_N x)\, \cos(\omega_N t + \phi_N),$$ (15.59)

corresponding to each possible value of the integer $N$. These are standing waves and each value of $N$ defines a different mode of the standing wave. The solution with $N=1$ is called the fundamental mode or first harmonic. We have,

(15.60)

which is shown in the left panel of Figure 15.9. The fundamental mode has wavelength $\lambda_1=2 L$ and frequency $\nu_1=c_s/2L$.

Figure 15.9: Standing wave modes: 1$^{st}$ and 2$^{st}$ harmonics

The second harmonic,

$$\xi(x,t)=A_2 \sin\left(\frac{2 \pi x}{L} \right) \, \cos \left(\frac{c_s 2 \pi t}{L} + \phi_2 \right),$$ (15.61)

which is shown in the right panel of Figure 15.9 has wavelength $\lambda_2=L$ and frequency $\nu_2=c_s/L$. The higher harmonics have wavelengths $\lambda_3=\lambda_1/3$, $\lambda_4= \lambda_1/4$, $\lambda_5= \lambda_1/5$ and frequencies $\nu_3=3 \nu_1$, $\nu_4=4 \nu_1$, respectively.

Each standing wave is a superposition of a left travelling and a right travelling wave. For example,

$$\sin(\omega_1 t + k_1 x) - \sin(\omega_1 t - k_1 x)= 2 \sin(k_1 x) \, \cos(\omega_1 t),$$ (15.62)

gives the fundamental mode. At all times the left travelling and write travelling waves exactly cancel at $x=0$ and $x=L$.

Figure 15.10: Arbitrary travelling wave in a string

Any arbitrary disturbance of the string can be expressed as a sum of standing waves

(15.63)

The resultant disturbance will in general not be a standing wave but will travel along the string as shown in Figure 15.10.

Question : If a string which is fixed at both the ends is plucked at an arbitrary point then which of the modes will not be excited?

Answer : Read about Young-Helmholtz law.

Problems

1. Consider the wave equations
   
   $$\left[ \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \right] \psi(\vec{r},t)=\frac{1}{L^2} \psi(\vec{r},t)$$
   
   

   a.

   What is the dispersion relation for this wave equation?

   b.

   Calculate the phase velocity and the group velocity.

   c.

   Analyze the behaviour when and $k \gg 1/L$

2. For the wave equation given below (where $c_s$ is a constant)
   
   $$\left[ 4 \frac{\partial^2}{\partial x^2}+ 9 \frac{\partial^2... ...c_s^2}\frac{\partial^2}{\partial t^2} \right] \xi(\vec{r},t)=0$$
   
   

   a.

   What is the speed of a travelling wave solution propagating along the $x$ axis?

   b.

   What is the speed of a travelling wave solution propagating along the $y$ axis?

   c.

   For what value of $b$ is the travelling wave given below a solution of the wave equation given above?
   

   $$\xi(\vec{r},t)=e^{-[b x + y - 5 c_s t]^2}$$

   
   

   d.

   What is the speed of the travelling wave given above ?

3. Consider the longitudinal disturbance
   
   $$\xi(x)=\frac{4}{4+x^2}$$
   
   
   inside an elastic rod where $c_s=2 {\rm m/s}$.

   a.

   Plot the disturbance as a function of $x$.

   b.

   What is $\xi(x,t)$ if the disturbance is a travelling wave moving along $+x$?

   c.

   Plot the disturbance as a function of $x$ at $t= 3~ {\rm s}$.

4. Which of the following are travelling waves? If yes, what is the speed?

   a.

   $\xi(x,t)=\sin^2 [ \pi (a x + b t)]$ b. $\xi(x,t)=\sin [ \pi (a x + b t)^2]$

   c.

   $\xi(x,t)=\sin^2 [ \pi (a x^2 + b t)]$ d. $\xi(x,t)=e^{-[a x^2 + b t^2 + 2 \sqrt{a b} x t]}$

   e.

   $\xi(\vec{r},t)=e^{-[a x + b y- t]^2/L^2}$

5. Consider a spherical wave $\xi(r,t)=a \sin[k(r-c_s t)]/r$ with $k=3,~{\rm m}^{-1}$ and $c_s=330 \, {\rm m s}^{-1}$.

   a.

   What is the frequency of this wave ?

   b.

   How much does the amplitude of this wave change over $\Delta r= 2 \pi/k$?

   c.

   In which direction does this wave propagate?

6. A longitudinal travelling wave
   
   $$\xi(x,t)= A e^{-\left[ x - c_s t \right]^2/L^2} \hspace{1 cm} [A=10^{-4} {\rm m}, \, L= 1 \, {\rm m}]$$
   
   
   passes through a long steel rod for which $\rho=8000 \, {\rm kg/m^3}$ and $Y=200 \times 10^9 \, {\rm N/m^2}$.

   a.

   At which point is the displacement maximum at $t=10^{-3} {\rm s}$?

   b.

   Plot the displacement at $x=5 \, {\rm m}$ as a function of time.

   c.

   Plot the velocity of the steel at $x=5 \, {\rm m}$ as a function of time.

   d.

   When is the velocity of the steel zero at $x=5 \, {\rm m}$?

7. Consider a longitudinal wave
   
   $$\xi(x,t)=A [\cos(\omega_1 t) \, \sin\left(\frac{\pi x}{L}\right) + \cos(\omega_2 t) \, \sin\left(\frac{2 \pi x}{L}\right)]$$
   
   
   in a steel rod of length $L=10 \, {\rm cm}$.

   a.

   What are the values of the angular frequencies $\omega_1$ and $\omega_2$ of the fundamental mode and the second harmonic respectively ?

   b.

   After what time period does the whole displacement profile repeat?

8. Consider a longitudinal standing wave
   
   $$\xi(x,t)=A \cos(\omega_1 t) \, \sin\left(\frac{\pi x}{L}\right) \hspace{1 cm} [A=10^{-4} {\rm m}]$$
   
   
   in a steel rod of length $L=5 \, {\rm cm}$.

   a.

   What is the instantaneous kinetic energy per unit volume at $x=2.5 {\rm cm}$?

   b.

   What is the instantaneous potential energy per unit volume at $x=2.5 {\rm cm}$?

   c.

   What is the time averaged kinetic energy per unit volume at ?

   d.

   What is the time averaged potential energy per unit volume at $x=2.5 {\rm cm}$?

9. Two steel rods, one $1 {\rm m}$ and another longer are both vibrating in the fundamental mode of longitudinal standing waves. What is the time period of the beats that will be produced?
10. Write the three dimensional Laplacian operator of equation (15.40) in spherical polar co-ordinates, and hence obtain the equation (15.41).