Wave Motion and String Waves Concept Page - 7

Example

Use principle of superposition to find total displacement at a given point

Example: Equations of two progressive waves at a certain point in a medium are given by  :
 y1=a(sinωt+θ1)$y_1=a(\sin \omega t+{\theta }_1)$ and
 y2=a(sinωt+θ2)$y_2=a(\sin \omega t+{\theta }_2)$.
If amplitude and time period of resultant wave formed by the superposition of these two waves are same as those of either wave, then find (θ1−θ2)$({\theta }_1-{\theta }_2)$

Solution:
y1=a sin(ωt+θ1)$y_1=asin(\omega t+{\theta }_1)$
y2=a sin(ωt+θ2)$y_2=asin(\omega t+{\theta }_2)$
y=y1+y2$y=y_1+y_2$
=a sin ωt cos θ1+a cos ωt sin θ1+a sin ωt cos θ2+a cos ωt sin θ2$=asin\omega tcos{\theta }_1+acos\omega tsin{\theta }_1+asin\omega tcos{\theta }_2+acos\omega tsin{\theta }_2$=a sin ωt(cos θ1+cos θ2)+a cos ωt(sin θ1+sin θ2)$=asin\omega t(cos{\theta }_1+cos{\theta }_2)+acos\omega t(sin{\theta }_1+sin{\theta }_2)$
So, for same amplitude and time period.
cos θ1+cos θ2=0$cos{\theta }_1+cos{\theta }_2=0$
and (sin θ1+sin θ2)max=1$(sin{\theta }_1+sin{\theta }_2{)}_{max}=1$
or, sin θ1+sin θ2=0$sin{\theta }_1+sin{\theta }_2=0$
and (cos θ1+cos θ2)max=1$(cos{\theta }_1+cos{\theta }_2{)}_{max}=1$
So, θ1−θ2=2π3.${\theta }_1-{\theta }_2={\displaystyle \frac{2\pi }{3}}.$

Example

Use principle of superposition to write the equation of resultant wave

Example: Standing waves are produced by the superposition of two waves: 
y1=0.05sin(3πt−2X)$y_1=0.05\sin (3\pi t-2_X)$ and
y2=0.05sin(3πt+2X)$y_2=0.05\sin (3\pi t+2_X)$, where x and y are expressed in metres and t is in seconds. Find the amplitude of a particle at x=0.5m$x=0.5m$ .
($($Given cos57.3∘=0.54)$\cos {57.3}^{\circ }=0.54)$

Solution:
y=y1+y2$y=y_1+y_2$ according to superposition of waves.
=0.05 sin (3πt−2x) + 0.05 sin(3πt+2x)$=0.05sin(3\pi t-2x)+0.05sin(3\pi t+2x)$
=2(0.05) cos 2x  sin 3πt$=2(0.05)cos2xsin3\pi t$
amplitude is 0.1 cos 2x$0.1cos2x$
at x=0.5$x=0.5$    A=0.1  cos 2(0.5)$A=0.1cos2(0.5)$
                           =0.1  cos 1radian$=0.1cos1radian$
                           =0.1  cos 57.3∘$=0.1cos{57.3}^{\circ }$
                           =0.1  × 0.54$=0.1\times 0.54$
                           =5.4 cm$=5.4cm$

Definition

Using superposition principle to draw the resultant of two superimposing waves

Example

Find the phase of resultant of two superimposing waves with same frequency

Example: Two waves of same amplitude and same frequency reach a point in a medium simultaneously. Find the phase difference between them for resultant amplitude to be zero.

Solution:
The equation of the two superimposing waves can be give by 
y1=asin(ωt)$y_1=a\sin (\omega t)$ and y2=asin(ωt+ϕ)$y_2=a\sin (\omega t+\varphi )$. 
So, the equation of the resultant wave is y=y1+y2=a(sin(ωt)+sin(ωt+ϕ))=0$y=y_1+y_2=a(\sin (\omega t)+\sin (\omega t+\varphi ))=0$
⇒sin(ωt)=−sin(ωt+ϕ)⇒ϕ=π,3π,...$\Rightarrow \sin (\omega t)=-\sin (\omega t+\varphi )\Rightarrow \varphi =\pi ,3\pi ,...$.

Definition

Constructive interference

Constructive interference: The interference of two or more waves of equal frequency and phase, resulting in their mutual reinforcement and producing a single amplitude equal to the sum of the amplitudes of the individual waves.
For Constructive interference phase difference between waves must be zero.

Example

Destructive interference

The interference of two waves of equal frequency and opposite phase, resulting in their cancellation where the negative displacement of one always coincides with the positive displacement of the other.

Condition for destructive interference is 180 degree phase difference between superimposing waves.

Definition

Destructive Interference

When two waves meet in such a way that their crests line up together, then it's called constructive interference. The resulting wave has a higher amplitude. In destructive interference, the crest of one wave meets the trough of another, and the result is a lower total amplitude.

Definition

Constructive Interference

Constructive Interference: the interference of two or more waves of equal frequency and phase, resulting in their mutual reinforcement and producing a single amplitude equal to the sum of the amplitudes of the individual waves.

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