Name: Oscillations of a Stretched String
Uploaded: 2019-07-19
Duration: 6 min 4 s
Description: In this video, we will learn about the standing wave formation in a string fixed at both ends and a few calculations on it for a better understanding.

Question

Briefly explain the concept of the superposition principle.

Accepted Answer

Superposition Principle: When a jerk is given to a stretched string that is tied at one end, a wave pulse is produced and the pulse travels along the string. Suppose two persons holding the stretched string on either side give a jerk simultaneously, then these two wave pulses move towards each other, meet at some point and move away from each other with their original identity. Their behavior is very different only at the crossing/meeting points; this behavior depends on whether the two pulses have the same or different shapes.
When the pulses have the same shape, at the crossing, the total displacement is the algebraic sum of their individual displacements and hence its net amplitude is higher than the amplitudes of the individual pulses. Whereas, if the two pulses have the same amplitude but shapes are 180° out of phase at the crossing point, the net amplitude vanishes at that point and the pulses will recover their identities after crossing.

Only waves can possess such a peculiar property and it is called superposition of waves. This means that the principle of superposition explains the net behaviour of the waves when they overlap. Generalizing to any number of waves i.e., if two are more waves in a medium move simultaneously, when they overlap, their total displacement is the vector sum of the individual displacements. We know that the waves satisfy the wave equation which is a linear second order homogeneous partial differential equation in both space coordinates and time. Hence, their linear combination (often called as linear superposition of waves) will also satisfy the same differential equation. To understand mathematically, let us consider two functions which characterize the displacement of the waves, for example,
$y 1 = A 1 \sin (k x - ω t) a n d y 2 = A 2 \cos (k x - ω t) S i n c e, b o t h y_{1} a n d y_{2} s a t i s f y t h e w a v e e q u a t i o n (s o l u t i o n s o f w a v e e q u a t i o n) t h e n t h e i r a l g e b r a i c s u m y = y 1 + y 2 a l s o s a t i s f i e s t h e w a v e e q u a t i o n . T h i s m e a n s, t h e d i s p l a c e m e n t s a r e a d d i t i v e . S u p p o s e w e m u l t i p l y y 1 a n d y 2 w i t h s o m e c o n s \tan t t h e n t h e i r a m p l i t u d e i s s c a l e d b y t h a t c o n s \tan t F u r t h e r, i f C_{1} a n d C_{2} a r e u s e d t o m u l t i p l y t h e d i s p l a c e m e n t s y_{1} a n d y_{2}, r e s p e c t i v e l y, t h e n, t h e i r n e t d i s p l a c e m e n t y i s C = C_{1} y_{1} + C_{2} y_{2} T h i s c a n b e g e n e r a l i z e d t o a n y n u m b e r o f w a v e s . I n t h e c a s e o f n s u c h w a v e s i n m o r e t h a n o n e d i m e n s i o n t h e d i s p l a c e m e n t s a r e w r i t t e n u \sin g v e c t o r n o t a t i o n . H e r e, t h e n e t d i s p l a c e m e n t y ⃗ i s \underset{y}{\to} = \sum_{i = 1}^{n} C_{i} \underset{y_{i}}{\to}$

The principle of superposition can explain the following :
(a) Space Interference
(b) Time Interference (also known as Beats)
(c) Concept of stationary waves
Waves that obey principle of superposition are called linear waves (amplitude is much smaller than their wavelengths). In general, if the amplitude of the wave is not small then they are called non-linear waves. These violate the linear superposition principle, e.g., laser. In this chapter, we will focus our attention only on linear waves.

Briefly explain the concept of the superposition principle.

Important Questions on Waves

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