Standing Sound Waves - Practice Questions & MCQ

Standing waves

When two sets of progressive wave of same type (both longitudinal or both transverse) having the same amplitude and same time period or frequency or wavelength travelling along the same straight line with same speed in opposite directions superimpose, a new set of waves are formed. These are called stationary waves.

Some of the characteristics of standing waves :

(1) In this the disturbance is confined to a particular region between the starting point and reflecting point of the wave.
(2) In this there is no forward motion of the disturbance from one particle to the adjoining particle and so on, beyond this particular region.
(3) The total energy in stationary wave is twice the energy of each of incident and reflected wave. But there is no flow or transference of energy along the stationary wave.
(4) Points in a standing wave, which are permanently at rest. These are called nodes. The distance between two consecutive nodes is $\frac{\lambda }{2}$

(5) The Points on the standing wave having maximum amplitude is known as antinodes. The distance between two consecutive antinodes is also $\frac{\lambda }{2}$
(6) All the particles execute simple harmonic motion about their mean position (except those are at nodes) with the same time period.

Note - In standing waves, if the amplitude of component waves are not equal. Resultant amplitude at nodes will not be zero. It will be minimum . Because of this, some energy will pass across nodes and waves will be partially standing.

Let us take an example to understand and derive equation of standing wave - 

Let us take a string and when a string is under tension and set into vibration, transverse harmonic waves propagate along its length. If the length of string is fixed, reflected waves will also exist. These incident and reflected waves will superimpose to produce transverse stationary waves in a string

                                                                    

Incident wave $y_1=a\sin \frac{2\pi }{\lambda }(vt+x)$
Reflected wave $y_2=a\sin \frac{2\pi }{\lambda }[(vt-x)+\pi ]=-a\sin \frac{2\pi }{\lambda }(vt-x)$
Now we can apply principle of superposition on this and get -

$y=y_1+y_2=2a\cos \frac{2\pi vt}{\lambda }\sin \frac{2\pi x}{\lambda }$

Standing Wave in a Closed Organ Pipe -
Organ pipes are the musical instrument which are used for producing musical sound by blowing air into the pipe. In this longitudinal stationary waves are formed due to superimposition of incident and reflected longitudinal waves.

A closed organ pipe is a cylindrical tube having an air column with one end closed. Sound waves are enters from a source vibrating near the open end. An ingoing pressure wave gets reflected from the fixed end. This inverted wave is again reflected at the open end. After two reflections, it moves towards the fixed end and interferes with the new wave sent by the source in that direction. The twice reflected wave has travelled an extra distance of $2l$ causing a phase advance of $\frac{2\pi }{\lambda }.2l=\frac{4\pi l}{\lambda }$
Similarly at open ends, the twice reflected wave suffered a phase change of $\pi$ at the open end.
So the phase difference is $\delta =\frac{4\pi l}{\lambda }+\pi$. Also the waves interfere constructively if phase difference is $2n\pi$

$\begin{array}{rl}& \frac{4\pi l}{\lambda }+\pi =2n\pi \\ & l=(2n-1)\frac{\lambda }{4}\end{array}$
 

Here $n=1,2,3\dots$. But if we take $n=0,1,2,\dots$ then the above equation can also be written as -

$l=(2n-1)\frac{\lambda }{4}$

So, the frequency can be written as -

$\nu =\frac{v}{\lambda }=\frac{v\cdot (2n-1)}{4l}$

Equation of standing wave is given by and explained earlier $=y=2a\cos \frac{2\pi t}{\lambda }\sin \frac{2\pi x}{\lambda }$
As, general formula for wavelength defined earlier $=\lambda =\frac{4L}{(2n-1)}$
The minimum allowed frequency is obtained by putting $\mathrm{n}=1$

                                                  

(1) First normal mode of vibration : $n_1=\frac{v}{4L}$

This is called fundamental frequency. The note so produced is called fundamental note or first harmonic.
(2) Second normal mode of vibration : $n_2=\frac{v}{{\lambda }_2}=\frac{3v}{4L}=3n_1$

This is called third harmonic or frst overtone.
(3) Third normal mode of vibration : $n_3=\frac{5v}{4L}=5n_1$

This is called fifth harmonic or second overtone.

Standing Waves in Open Organ Pipes
General formula for wavelength -

$\lambda =\frac{2L}{n}\text{ where }n=1,2,3\dots \dots \dots$

Then the first normal mode of vibration is -

$n_1=\frac{v}{{\lambda }_1}=\frac{v}{2L}$
 

This is called fundamental frequency and the node so produced is called fundamental node or first harmonic.

 

(2) Second normal mode of vibration $n_2=\frac{v}{{\lambda }_2}=\frac{v}{L}=2\left(\frac{v}{2L}\right)=2n_1\Rightarrow n_2=2n_1$

This is called second harmonic or first overtone.
(3) Third normal mode of vibration $n_3=\frac{v}{{\lambda }_3}=\frac{3v}{2L},n_3=3n_1$

This is called third harmonic or second overtone.