Vibration of strings

Standing waves also known as stationary waves, can be excited in stretched string or in air columns (organ pipes). Later we will see that because of the boundary conditions a vibrating system can oscillate in more than one frequencies which are called their naturals frequencies.

These frequencies depend on the length of the system (or stretched string or organ pipes), boundary conditions and the wave speed in the medium.

Now suppose some periodic driving force is applied on the vibrating system.

After some initial the system finally oscillates with the frequency of the driving force, and at resonance (i.e. when frequency of the driving force is equal to any one of the natural frequency of the system) the amplitude of the oscillations become too large.

When we bring a vibrating tuning fork near the mouth of an organ pipe, then at resonance (i.e. when the length of the organ pipe is so adjusted that its frequency corresponding to that is equal to the frequency of the tuning fork) the air column in the organ pipes starts oscillating with great amplitude.

Similarly as a string has many natural frequencies, so when it is excited with a tuning fork (or a vibrating body) the string will be in resonance with the given body if any one of its natural frequency coincides with that of the body.

Stationary Waves with Rigid Boundaries at both ends :

For Pth harmonic , Total number of nodes between the two fixed ends = (P + 1)

Distance between two nodes = λ/2

$ \displaystyle l = P\frac{\lambda}{2} $

$ \displaystyle \lambda = \frac{2 l}{P} $

Now frequency n = v/λ    where v is the speed of wave in the string, $ \displaystyle v = \sqrt{\frac{T}{m}} $

$ \displaystyle n = \frac{P}{2l}v $

$ \displaystyle n = \frac{P}{2l}\sqrt{\frac{T}{m}} $

P = 1 , 2 , 3 , …

For P = 1 ,

$ \displaystyle n_1 = \frac{1}{2l}\sqrt{\frac{T}{m}} $ = Fundamental frequency or frequency of Ist harmonic.

For P = 2 ,

$ \displaystyle n_2 = \frac{2}{2l}\sqrt{\frac{T}{m}} $ = 2n1 = frequency of 2nd harmonic

From this we find that- n1: n2: n3 = 1:2:3:

i.e. in case of a string stretched at both ends, we get both even and odd harmonics.

Also Read :

→ Wave & Wave Function
→ progressive wave , Equation of a Plane Progressive Wave
→ Speed of Wave in a String
→ Sound Waves , Infrasonic & Ultrasonic Wave
→ Velocity of Sound : Newton’s Formula & Laplace’s correction
→ Factor affecting Velocity of Sound
→ Displacement wave & Pressure wave
→ Pitch, Loudness & Quality of Sound Wave
→ Reflection, Refraction & Superposition of Waves
→ Standing Waves in a String
→ Vibration of air columns : Closed & Open organ Pipe
→ Beats
→ Doppler Effect

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