physics-class-11-unit-15-chapter-03-reflection-of-waves-standing-waves-ona-string-l-3-5-xg-9uhw34h4

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Waves </primary>
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- <span style="color:green">Dispersionless waves</span>

* $f(x, t)=f(x-v t)$
 
* $f(t-v / v)\quad$  waves travelling in the positive x direction

- For the wave travelling in the $-x$ direction

- $f(x, t) =  f(x+v t) $

- $f(t+x / v)$

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<footer style = "font-size: 18px"> $\rightarrow$ Reflection of Waves, Superposition of Waves, Standing Waves on a String and Their Frequencies $\rightarrow$ <span style = "color:blue"> Waves </span> $\rightarrow$ Waves $\rightarrow$ SUPERPOSITION OF WAVES </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> SUPERPOSITION OF WAVES </primary>
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- If there are two or more waves arriving at a point at the same time

- $f_1(x, t), \quad f_2(x, t)+\cdots$

- Then the net displacement / pressure at that point is given by the sum of individual displacement/pressure.

- $f(x, t)=f_1(x, t)+f_2(x, t)+\cdots$

- Waves satisfying linear differential  equation
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<footer style = "font-size: 18px">Waves $\rightarrow$ Waves $\rightarrow$ <span style = "color:blue"> SUPERPOSITION OF WAVES </span> $\rightarrow$ Principle of Superposition $\rightarrow$ Consequances of superposition principle </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Principle of Superposition </primary>
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- $f(x, t)=f_1(x, t)+f_2(x, t)+\ldots$

- <span style="color:blue"><ins>Specialize in sinusoidal waves</ins></span>

- **$f(x,t)=$**

- A $ \sin (k x-\omega t)\rightarrow$,  A $\sin (\omega t-k x)\rightarrow$

- A $G_s(k x-\omega t)\rightarrow$,  A $\sin (k x+\omega t)\leftarrow$

- <span style="color:green"> $k=\frac{\omega}{v}, \quad k=\frac{2 \pi}{\lambda}, \quad v=2 \lambda $</span>

- $f(x, t)=  A_1 \sin  (k_1 x-\omega_1 t) $

- $+A_2 \sin (k_2 x-\omega_2 t)$,  $ +A_3 \sin (k_2 x+\omega_3 t)$, and $ +B_1 \cos (k_4 x+\omega_4 t)$

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![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-15-chapter-03-reflection-of-waves-standing-waves-ona-string-l-3_5-xg_9uhw34h4-04.jpg)

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<footer style = "font-size: 18px">Waves $\rightarrow$ SUPERPOSITION OF WAVES $\rightarrow$ <span style = "color:blue"> Principle of Superposition </span> $\rightarrow$ Consequances of superposition principle $\rightarrow$ Reflection of waves at a boundary </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Reflection of waves at a boundary </primary>
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- x = 0 by a hard wall

- displacement at the point when the string is tied = 0

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<footer style = "font-size: 18px">Principle of Superposition $\rightarrow$ Consequances of superposition principle $\rightarrow$ <span style = "color:blue"> Reflection of waves at a boundary </span> $\rightarrow$ Reflection of a Wave on a String at the Point Where the String is tied $\rightarrow$ A Sinusoidal Wave </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Reflection of a Wave on a String at the Point Where the String is tied </primary>
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- At the boundary at  x = 0

- Net displacement = 0

- Net displacement = Sum of individual displacement 
- = displacement due to incoming wave + displacement due to the reflected wave

- 0 = displacement due to in coming wave + displacement due to the reflected wave

- $\Rightarrow $ Displacement of the reflaction wave = -( displacement due to the incoming wave)

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<footer style = "font-size: 18px">Consequances of superposition principle $\rightarrow$ Reflection of waves at a boundary $\rightarrow$ <span style = "color:blue"> Reflection of a Wave on a String at the Point Where the String is tied </span> $\rightarrow$ A Sinusoidal Wave $\rightarrow$ A sinusoidal wave </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> A Sinusoidal Wave </primary>
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- <span style="color:green">Incoming wave</span>

- $ y_{incoming}(x, t)=A \sin (k x-\omega t)$

- $ y_{\text {reflecter }}(x, t)=B \sin (k x+\omega t)$

- **We are looking at $x=0$**

- $y_{\text {net }}(x=0)=-A \sin \omega t+B \sin \omega t=0 \Longrightarrow B=A$

- In coming wave $=A \sin (k x-\omega t)$

- Reflector wave = A Sin $(k x+\omega t)$

- Reflected wave at $x=0$

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<footer style = "font-size: 18px">Reflection of waves at a boundary $\rightarrow$ Reflection of a Wave on a String at the Point Where the String is tied $\rightarrow$ <span style = "color:blue"> A Sinusoidal Wave </span> $\rightarrow$ A sinusoidal wave $\rightarrow$ Reflection at Boundaries Creats Reflected Waves </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> A sinusoidal wave </primary>
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- <span style="color:green">Incoming wave</span>

- $A \sin \omega t  =-(-A \sin \omega t) =-y_{incoming}$

- $y_{reflected } =A \sin \omega t =A \sin (-\omega t+\pi) $

- $ A \sin (-\omega t+\pi) =-A \sin (-\omega t) -A \sin \omega t $

- $ y_{refected } =A \sin (-\omega t+\pi)$

- To get the reflected wave, we add a phase of $\pi$ to the phase of incoming wave

- <span style="color:green">$\Rightarrow$ There is a phase difference of $\pi$ between phase of incoming and reflected wave</span>

- The represents not a travelling wave but it represents a Standing Wave

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<footer style = "font-size: 18px">Reflection of a Wave on a String at the Point Where the String is tied $\rightarrow$ A Sinusoidal Wave $\rightarrow$ <span style = "color:blue"> A sinusoidal wave </span> $\rightarrow$ Reflection at Boundaries Creats Reflected Waves $\rightarrow$ Reflected Waves </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Reflection at Boundaries Creats Reflected Waves </primary>
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- <span style="color:blue">What happens to the displacement at the points?</span>

- <span style="color:green">Superposition of incoming and reflected waves gives rise to standing waves</span>
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<footer style = "font-size: 18px">A Sinusoidal Wave $\rightarrow$ A sinusoidal wave $\rightarrow$ <span style = "color:blue"> Reflection at Boundaries Creats Reflected Waves </span> $\rightarrow$ Reflected Waves $\rightarrow$ Standing Waves </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Reflected Waves </primary>
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- <span style="color:green">Net displacement at the boundary $\cancel=$ 0</span>

- <span style="color:blue">The $\Delta P$ of the boundary  $\cancel=$0</span>
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<footer style = "font-size: 18px">A sinusoidal wave $\rightarrow$ Reflection at Boundaries Creats Reflected Waves $\rightarrow$ <span style = "color:blue"> Reflected Waves </span> $\rightarrow$ Standing Waves $\rightarrow$ Standing Waves </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Standing Waves </primary>
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- A wave moving in the + x direction

- $A \sin (k x-\omega t)$

- Superpose this with a wave travelling in the - x direction

- $A \sin (k x+\omega t)$

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<footer style = "font-size: 18px">Reflection at Boundaries Creats Reflected Waves $\rightarrow$ Reflected Waves $\rightarrow$ <span style = "color:blue"> Standing Waves </span> $\rightarrow$ Standing Waves $\rightarrow$ Standing Wave on a String </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Standing Waves </primary>
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- $y(x, t)=  A \sin (k x-\omega t)+A \sin (k x+\omega t)=A \sin k x \cos \omega t-A \cos k x \sin \omega t+A \sin k x \cos \omega t+A \cos k x \sin \omega t$

- $y(x, t)=  2 A \sin k x \cos \omega t$

- This is not of the from $f(x-v)$

- $f(x+v t)$ - The represents not  a travelling wave  but it represents a **STANDING WAVE**

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<footer style = "font-size: 18px">Reflected Waves $\rightarrow$ Standing Waves $\rightarrow$ <span style = "color:blue"> Standing Waves </span> $\rightarrow$ Standing Wave on a String $\rightarrow$ Standing Wave on a String </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Standing Wave on a String </primary>
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- y(x,t) = (2A) $\sin k x \cos \omega t B\sin k x \cos \omega t$
 
- $ y(x, t)  =B \sin k x \cos \omega t=C \cos k x \text { Cost }=D \sin k x \sin \omega t$

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<footer style = "font-size: 18px">Standing Waves $\rightarrow$ Standing Waves $\rightarrow$ <span style = "color:blue"> Standing Wave on a String </span> $\rightarrow$ Standing Wave on a String $\rightarrow$ Standing Wave on a String </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Standing Wave on a String </primary>
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- **Different example of standing wave**

- $y(x, t)  =A \sin k x \cos \omega t=A \cos k x \sin \omega t$

- <span style="color:green">(1) Standing wave on a string</span>

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<footer style = "font-size: 18px">Standing Waves $\rightarrow$ Standing Wave on a String $\rightarrow$ <span style = "color:blue"> Standing Wave on a String </span> $\rightarrow$ Standing Wave on a String $\rightarrow$ Standing Wave on a String </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Standing Wave on a String </primary>
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- (1) String tied at Both Ends

- <span style="color:blue">$y(x, t)=A \sin k x \cos \omega t$</span>

- Length of the string is $-L$

- <span style="color:green">$ y(x, t)=A \sin k x \cos \omega t $</span>

- <span style="color:green">$ \text { At } x=0 \quad y=0 $</span>

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<footer style = "font-size: 18px">Standing Wave on a String $\rightarrow$ Standing Wave on a String $\rightarrow$ <span style = "color:blue"> Standing Wave on a String </span> $\rightarrow$ Standing Wave on a String $\rightarrow$ Standing Wave on a String </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Standing Wave on a String </primary>
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- (1) String tied at Both Ends

- <span style="color:blue">$y(x, t)=A \sin k x \cos \omega t$</span>

- Length of the string is $-L$

- <span style="color:green">$ y(x, t)=A \sin k x \cos \omega t $</span>

- <span style="color:green">$ \text { At } x=0 \quad y=0 $</span>

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### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Standing Wave on a String </primary>
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- $ \omega =2\pi\nu$

- $\nu_n = n(\frac{v}{2 L}\nu_n  =n(\frac{v}{2 L})=\frac{n}{2 L} \sqrt{\frac{T}{\mu}}\nu_n  =\frac{n}{2 L} \sqrt{\frac{T}{\mu}} $

- $\text { SinkL }=0$

- $ k  =\frac{2 \pi}{\lambda} L=n \pi\lambda=(\frac{2 L}{n})\frac{\lambda}{2} \times n=L \Rightarrow \lambda=\left(\frac{2 L}{n}\right)$

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<footer style = "font-size: 18px">Standing Wave on a String $\rightarrow$ Standing Wave on a String $\rightarrow$ <span style = "color:blue"> Standing Wave on a String </span> $\rightarrow$ Reflection of a wave $\rightarrow$ Reflection of a wave </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Reflection of a wave </primary>
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- Distance between nods $=\frac{\lambda}{2}$

- Distance detween two antinods $=\frac{\lambda}{2}$

- $\nu_n=\frac{n}{2 L} \sqrt{\frac{T}{\mu}}$

- $y(x, t)=A \sin k x \cos \omega t $

- $y(0, t)=0 $

- $y(x=L, t)=A \sin k L \cos \omega t$

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<footer style = "font-size: 18px">Standing Wave on a String $\rightarrow$ Standing Wave on a String $\rightarrow$ <span style = "color:blue"> Reflection of a wave </span> $\rightarrow$ Reflection of a wave $\rightarrow$ Reflection of a wave </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Reflection of a wave </primary>
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- <span style="color:green">maximum displacement at $x=L$</span>

- $k L=(2 n+1) \frac{\pi}{2}$

- $k=\frac{2 \pi}{\lambda}=\frac{(2 n+1) \pi}{2 L}$

- $\lambda=\frac{4 L}{(2 n+1)}$

- $\nu_n=\frac{v}{\lambda}  =\frac{v}{4 L / 2 n+1}=\frac{v}{4 L}(2 n+1) =\left(n+\frac{1}{2}\right) \cdot \frac{v}{2 L}\frac{\lambda}{4}(2 n+1)=L\Rightarrow \lambda=\left(\frac{4 L}{2 n+1}\right)$

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<footer style = "font-size: 18px">Standing Wave on a String $\rightarrow$ Reflection of a wave $\rightarrow$ <span style = "color:blue"> Reflection of a wave </span> $\rightarrow$ Reflection of a wave $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Reflection Of Waves Standing Waves Ona String L-3 </Success>
### <primary style="font-size:24px"> Reflection of a wave </primary>
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- <span style="color:blue">Conclude</span>

- (i) Principle of superposition

- $f(x, y)=\sum_i f_i(x, t)$

- (ii) Reflection of a wave from a boundary phase difference of $\pi$ between the phase

- of incoming and reflected waves when the boundary is hard $\Rightarrow$ No displacement at the boundary

- (III) Standing waves: Standing's wars on a string.

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<footer style = "font-size: 18px">Reflection of a wave $\rightarrow$ Reflection of a wave $\rightarrow$ <span style = "color:blue"> Reflection of a wave </span> $\rightarrow$ Thank You $\rightarrow$  </footer>