What is the principle of superposition of waves?
The principle of superposition of waves states that if two or more progressive waves traveling together in a medium, converges to a point, the resulting displacement of the particle at the point is equal to the algebraic (vector) sum of individual displacement of the waves. Let y1, y2, y3, ..., yn be the displacement at a point due to individual displacement, y at the same time when the waves superpose to each other is given by
y = y1 + y2 + y3 + .... + yn
Stationary (Or standing) wave: Whenever two progressive waves of the same wavelength and amplitude travel in opposite directions with the same speed, they are superimposed; a single wave is formed such wave is called a stationary wave.
This is formed as given by the superposition of principle which states that the vector sum of displacements of individual waves is equal to the displacement of the resultant wave. When a stationary wave is formed due to the superposition of two waves, the points of maximum and zero amplitude result alternatively in space. The points where the amplitude of vibration is maximum are called the antinodes and those where the amplitude is zero are called nodes. The distance between two consecutive nodes or antinodes is equal to half of the wavelength i.e., \(\frac12\lambda\), where \(\lambda\) is the wavelength of a wave. Also, the distance between adjacent nodes is equal to one-quarter of the wavelength i.e., \(\frac14\lambda\).
Stationary wave equation: Stationary wave equation can be obtained by adding vectorically the displacements of two waves of equal amplitude, frequency (or period), and wavelength travelling in opposite directions.
Let y1 be the displacement of the wave travelling to the positive x-direction, then,
\(y_1=a\sin\left(\omega t-kx\right)\) ---------- (i)
And, y2 be the displacement of the wave travelling to the negative x-direction. Then
\(y_2=a\sin\left(\omega t+kx\right)\) ---------- (ii)
Now by using the principle of superposition of waves, the resultant displacement y is given by
y = y1 + y2
= \(a\sin\left(\omega t-kx\right)+a\sin\left(\omega t+kx\right)\)
= \(a\left[\sin\left(\omega t-kx\right)+\sin\left(\omega t+kx\right)\right]\)
= \(a\left[\sin\left(\omega t\right)\cos\left(kx\right)-\cos\left(\omega t\right)\sin\left(kx\right)+\sin\left(\omega t\right)\cos\left(kx\right)+\cos\left(\omega t\right)\sin\left(kx\right)\right]\)
= \(2a\sin\left(\omega t\right)\cos\left(kx\right)\)
= \(2a\sin\left(\frac{2\pi}Tt\right)\cos\left(\frac{2\pi}\lambda x\right)\)
∴ \(y=A\sin\left(\frac{2\pi}Tt\right)\) ----------- (iii)
Where, \(A=2a\cos\left(\frac{2\pi x}\lambda\right)\) be the amplitude of the resultant wave.
Equation (iii) is the equation of a stationary wave. If v is the velocity of sound in air and l is the length of the organ pipe, then the fundamental frequency in an open organ pipe is
\(f_0=\frac v{2l}\) -------- (i)
Also, the fundamental frequency produced in a closed organ pipe whose length is the same as an open organ pipe is
\(f_c=\frac v{4l}\) --------- (ii)
∴ Dividing Equation (i) by (ii), we get
\(f_c=\frac12f_0\)
Hence, the frequency of the fundamental mode of a closed organ pipe is half as compared to that of an open pipe of the same length.