- All we know that waves are nothing but the disturbances produced which transfer energy without changing the position of the particles of the medium.
- When two identical waves which may be transverse or longitudinal traveling in opposite directions superimpose with each other then due to superposition the resultant wave produced is in the form of loops and such wave is called as stationary wave as shown in following figure.
- In case of stationary waves, the point at which displacement of particle is zero are called as nodes.
- And the points on the stationary wave at which displacement of the particle is maximum such points are called as antinodes.
- The nodes and antinodes are occurring alternately as shown in figure below.
- The distance between two successive antinodes or nodes becomes half of the wavelength of stationary wave.
- If we pluck the string at any point which is stretched by the rigid support then nodes are produced at the ending and antinodes is produced at the centre of the wave. Such mode of wave is called as the fundamental mode of the wave.
- And the corresponding frequency of the wave is called as fundamental frequency of the wave.
Laws of Vibrating String:
Statement:
If n is the frequency of the wave and ¥ is the wavelength of the wave then we can write the frequency of the wave as
n = 1/2L √T/m
Where, n is the frequency of the wave
L is the vibrating length
T is the tension in the string
m is the linear density of the string
1.) Law of length :
The law of length states that the fundamental frequency of the stretched string is inversely proportional to the vibrating length only when the tension T in the string and the linear density m of the string are become constant.
That means,
T α 1/L when T and m are constant
2.) Law of tension:
Law of tension states that the fundamental frequency of transverse vibration of stretched string is directly proportional to the square root of the tension T in the string only when linear density m and vibrating length L of the string become constant.
That means,
n α √T when L and m are constant.
3.) Law of density:
The law of density states that, the fundamental frequency of transverse vibration of stretched string is inversely proportional to the square root of the linear density m of the wire only when vibrating length L and tension T in the string are kept constant.
That means,
n α 1/√m when T and L are constant.
Example:
If the two wires which are identical in nature are stretched by 25 kg and 9kg weight respectively. And if resonating length for first wire was found to be 20cm then what will be the resonating length for the second wire.
Solution:
Given that,
For first wire, T1 = 25 kg and L1 = 20cm
For second wire, T2 = 9 kg and L2 = ?
We have from the law of tension,
√T1/√T2 = L1/L2
Thus, √25/√9 = 20/L2
Thus, 5/3 = 20/L2
And hence, L2 = 20*3/5= 12cm
Thus, the resonating length for the second wire was found to be 12 cm.