Question

How to find the particle velocity when eqation of standing wave is given?

Answer 1

When two progressive waves of same amplitude and wavelength travelling along a straight line in opposite directions superimpose on each other, stationary waves are formed.

Analytical method

Let us consider a progressive wave of amplitude a and wavelength λ travelling in the direction of X axis.

y1 = a sin 2π [t/T – x/λ]         …... (1)

This wave is reflected from a free end and it travels in the negative direction of X axis, then

y2 = a sin 2π [t/T + x/λ]         …... (2)

According to principle of superposition, the resultant displacement is,



y = y1+y2

= a [sin 2π (t/T – x/λ) + sin 2π (t/T + x/λ)]

= a [2sin (2πt/T) cos  (2πx/λ)]

So, y = 2a cos (2πx/λ) sin (2πt/T)         …... (3)

This is the equation of a stationary wave.

(a) At points where x = 0, λ/2, λ, 3λ/2, the values of cos 2πx/λ = ±1

∴ A = + 2a. At these points the resultant amplitude is maximum. They are called antinodes as shown in figure.

(b) At points where x = λ/4, 3λ/4, 5λ/4..... the values of cos 2πx/λ = 0.

∴ A = 0. The resultant amplitude is zero at these points. They are called nodes.



The  distance  between  any  two successive antinodes or nodes is equal to λ/2 and the distance between an antinode and a node is λ/4.

(c) When t = 0, T/2, T, 3T/2, 2T,.... then sin 2πt/T = 0, the displacement is zero.

(d) When t = T/4, 3T/4, 5T/4 etc,....sin 2πt/T = ±1,  the displacement is maximum.

hope it helps!!

Answer 2

When two progressive waves of same amplitude and wavelength travelling along a straight line in opposite directions superimpose on each other, stationary waves are formed.

Analytical method

Let us consider a progressive wave of amplitude a and wavelength λ travelling in the direction of X axis.

y1 = a sin 2π [t/T – x/λ]         …... (1)

This wave is reflected from a free end and it travels in the negative direction of X axis, then

y2 = a sin 2π [t/T + x/λ]         …... (2)

According to principle of superposition, the resultant displacement is,



y = y1+y2

= a [sin 2π (t/T – x/λ) + sin 2π (t/T + x/λ)]

= a [2sin (2πt/T) cos  (2πx/λ)]

So, y = 2a cos (2πx/λ) sin (2πt/T)         …... (3)

This is the equation of a stationary wave.

(a) At points where x = 0, λ/2, λ, 3λ/2, the values of cos 2πx/λ = ±1

∴ A = + 2a. At these points the resultant amplitude is maximum. They are called antinodes as shown in figure.

(b) At points where x = λ/4, 3λ/4, 5λ/4..... the values of cos 2πx/λ = 0.

∴ A = 0. The resultant amplitude is zero at these points. They are called nodes.



The  distance  between  any  two successive antinodes or nodes is equal to λ/2 and the distance between an antinode and a node is λ/4.

(c) When t = 0, T/2, T, 3T/2, 2T,.... then sin 2πt/T = 0, the displacement is zero.

(d) When t = T/4, 3T/4, 5T/4 etc,....sin 2πt/T = ±1,  the displacement is maximum.

hope it helps!!