Stationary Waves (Standing Waves)

 

Definition:     

        "When two plane waves having the same amplitude and frequency, travelling with the same speed in opposite direction along a line, are superposed, a wave obtained is called stationary or standing wave".

Explanation:

The sound produced by most of string and wind musical instruments is due to the formation of stationary waves or standing waves in these instruments. The vibration in the string of a guitar or piano set up stationary waves of definite frequencies.

Stationary waves cannot be set up in any medium which do not transmit energy from one place to another place like progressive waves.

Characteristics of Stationary Waves

  1. No net energy is transferred from particle to particle in standing waves.
  2. All particles except from nodes perform S.H.M.
  3. Node:     "The points of zero displacement on stationary waves are called nodes and denoted by N".
  4. Anti-node:     "The points of maximum displacement on stationary waves are called anti-nodes and denoted by A".
  5. Wavelength:     The distance between two successive nodes or anti-nodes is equal to half of the wavelength (λ/2), while the distance between adjacent node and anti-node is equal to (λ/4).
  6. At node strain and pressure are maximum and displacement is zero but at antinode, displacement is maximum and strain and pressure are at minimum.

Relation between frequency and wavelength

As we know that by increasing the wiggling frequency, the number of loops increases. As a result, wavelength of the wave decreases. However, the product of the frequency and wavelength is always equal to the velocity of waves.

Also, the relation between speed, frequency and wavelength is mathematically given by;

V = fλ


Since for the same cord, velocity remains constant. Therefore;

The above expression shows that wavelength is inversely proportional to frequency. Thus by increasing frequency of the waves, waves decrease.

Fundamental and overtone vibration

Definition:     The characteristic frequency of vibration f1, is called the fundamental frequency and the higher frequencies f2, f3, f4, …..  which are the integral multiples of the fundamental are called overtones or harmonies. Explanation:

Suppose the system is capable of vibrating at number of frequencies f1, f2, f3, …... f1 < f2 < f3 …... so that Lowest frequency f1 is called the fundamental frequency and mode of oscillation is called the fundamental mode. The higher frequencies are called overtones with f2 being the first overtones. f3 the second overtone, and so on.

The tones are the integral multiples of the fundamental (in certain systems) fn=nf1. where n=1 ,2,3,4 …...

The overtones are called harmonics. The first member of the harmonics sequence is the fundamental; the second harmonic is the first overtone and so on.

Transverse Stationary Waves in a Stretched String

Definition:         "A standing wave obtained due to the superposition of transverse waves is called transverse stationary wave".

Demonstration: To demonstrate mechanical transverse stationary waves, consider a string of length "L", which is kept stretched by clamping its two ends so that the tension in the string is "T" as shown in the figure. To find the characteristics frequencies of vibration, we have to pluck the string at different places.

String Plucked at its Middle

When the string is plucked at its middle point and then released, two waves are generated which moves in opposite directions along the string. Both of these are reflected back from the clamped ends. They meet at the middle where they superpose each other and as a resultant a stationary wave is setup. The whole string will vibrate in one loop, with nodes at the fixed ends and anti-node at the middle as shown in the figure (a). 

The frequency of the stationary wave is equal to the frequency of the two progressive waves f1. To establish a relationship between the length of the string and wavelength λ, of the waves we know that the distance between successive nodes is equal to half a wavelength, i.e.

      ⇒   (1)
From the relation between speed, frequency and wavelength, we have;

Putting the value of λ1, from equation (1), we get;

         ⇒   (2) 

If M is the total mass of the string, then the speed V of the progressive wave along the string is given by;

Where T is the tension in the string and L is the length. So the frequency f1 is;

       ⇒   (3)

This characteristic frequency f1 of vibration is called the fundamental frequency or first harmonic.

String plucked at quarter length

When the string is plucked at one quarter of its length (L/4), again standing wave will setup. But this time the string will vibrate in two loops as shown in figure (b).

Let λ2 is the wavelength and ‘f2’ is the frequency in this mode of vibration. Now;



        ⇒   (4)

We Know that,


Putting the value of λ2 form equation (4), we get;


              

       ⇒   (5)

String plucked at one sixth of its length

When the string id plucked at one sixth of its length (L/6), it will vibrate in three loops, having three anti-nodes and four nodes as shown in figure (c). 

So,



     ⇒  (6)  

We know that,

  

Putting the value of λ3 from equation (6), we get;

               

     ⇒  (7)

String plucked at arbitrary point (n)

To generalize the above discussion, let the string is plucked at some arbitrary point, so that the string vibrates in ‘n’ number of loops, with (n+1) nodes and nth anti-nodes.

Now from equation (4) and equation (6), we have;







                      (For n-number of loops)

This gives the general expression for wavelength of standing waves in a medium, vibrating in any arbitrary number of loops.

And the frequency ‘fn’ is;

      

Putting the value of 𝛌n, we get;



     ⇒   (8)

Quantization of frequencies:

It is found that in any medium stationary waves of all frequencies cannot be set up. The waves having a discrete set of frequencies only, can be set up in the medium. This fact is known as the quantization of frequencies, nf1, 

where n=1, 2, 3, 4 …...

This also means that in a medium, standing waves can be produced only in discrete order of frequencies, i.e. f1, 2f1, 3f1, ...

The lowest of these frequencies 'f1' is called fundamental frequency or first harmonic. The higher frequencies, e.g. 'f2', 'f3', ‘f4’ etc. are called overtones.