Question

Consider a string fixed at one end. A travelling wave given by the wave equation $y = A \sin (ω t - k x)$ is incident on it.

Show that at the fixed end of a string the wave suffers a phase change of $π,$ i.e., as it travels back as if the wave is inverted.

Accepted Answer

Wave equation of incident wave is

$y_{1} = A \sin (ω t - k x)$ (positive $x$ -direction)

When the wave strikes the fixed end it must be reflected. The wave will now be travelling in the negative $x$ -direction, and so its equation is

$y_{2} = A \sin (ω t + k x + ϕ)$ (negative $x$ -direction)

The phase constant has been added to account for any phase change after reflection. Let us take $x = 0$ 0$ at the fixed end. The behaviour of a wave at particular positions is governed by appropriate boundary conditions. For fixed ends, the boundary condition is that the end point is a node.

The resultant motion due to incident and reflected wave is

$y = y_{1} + y_{2}$
$y = A [\sin (ω t - k x) + \sin (ω t + k x + ϕ)] . . . (i)$

The boundary condition that must be satisfied by the resultant wave on string is $y = 0$ at $x = 0$

On substituting these values in Eq. $(i),$ we obtain

$\sin ω t + \sin (ω t + ϕ) = 0$
$\sin ω t = - \sin (ω t + ϕ) . . . (ii)$

Equation $(ii)$ must be satisfied at all times. For the sake of convenience, we take $ω t = π / 2 .$

$\sin (\frac{π}{2} + ϕ) = - 1$

$\sin θ = - 1$ implies $θ$ can have any of the following values.

$3 π / 2, 7 π / 2, 11 π / 2, \dots$

Therefore $ϕ$ can have any of the values $π, 3 π, 5 π$ etc. All these values are physically possible and distinguishable. We choose the simplest one, so $ϕ = π .$

B M Sharma Solutions for Chapter: Superposition and Standing Waves, Exercise 1: ILLUSTRATIONS

B M Sharma Physics Solutions for Exercise - B M Sharma Solutions for Chapter: Superposition and Standing Waves, Exercise 1: ILLUSTRATIONS

Questions from B M Sharma Solutions for Chapter: Superposition and Standing Waves, Exercise 1: ILLUSTRATIONS with Hints & Solutions

Learn and Prepare for any exam you want !