physics-class-11-unit-15-chapter-04-standing-wavesina-pipe-phenomena-of-beats-doppler-effect-l-4-5-mhskub1knc0

### <Success style="font-size:25px"> Standing Wavesina Pipe Phenomena Of Beats Doppler Effect L-4 </Success>
### <primary style="font-size:24px"> Superposition of Waves </primary>
<hr>
<main>

- <span style="color:green">Standing wave</span>

- Discussed Standing waves on a string
</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px"> $\rightarrow$ Standing Waves in a Pipe Phenomena of Beats Doppler Effect $\rightarrow$ <span style = "color:blue"> Superposition of Waves </span> $\rightarrow$ Vibrations of an Air Column in a Pipe $\rightarrow$ Vibrations of an Air Column in a Pipe </footer>

### <Success style="font-size:25px"> Standing Wavesina Pipe Phenomena Of Beats Doppler Effect L-4 </Success>
### <primary style="font-size:24px"> Vibrations of an Air Column in a Pipe </primary>
<hr>
<main>

- Vibration of an air column is described by pressure variation

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Standing Waves in a Pipe Phenomena of Beats Doppler Effect $\rightarrow$ Superposition of Waves $\rightarrow$ <span style = "color:blue"> Vibrations of an Air Column in a Pipe </span> $\rightarrow$ Vibrations of an Air Column in a Pipe $\rightarrow$ Vibrations of an Air Column in a Pipe </footer>

- $\Delta p=A \operatorname{Sin} k x \operatorname{Cos} \omega t 
\Rightarrow \Delta p(x=0)=0$

- $\Delta p(x=L)=0 \Rightarrow \operatorname{Sin} k L=0$

- $ k=\frac{n \pi}{L}$
</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Superposition of Waves $\rightarrow$ Vibrations of an Air Column in a Pipe $\rightarrow$ <span style = "color:blue"> Vibrations of an Air Column in a Pipe </span> $\rightarrow$ Vibrations of an Air Column in a Pipe $\rightarrow$ Vibrations of an Air Column in a Pipe </footer>

- $\Delta p(x, t)=A \operatorname{Sin} k x \operatorname{Cos} \omega t $

- $ \Delta p(x=L, t)=0 \quad \operatorname{SinkL}=0 $

- $ k=\frac{n \pi}{L}$

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

- $ \nu_n=\frac{v}{\lambda}=n\left(\frac{v}{2 L}\right)$

- $ \nu_1=\frac{v}{2 L}, $
 
- $ \nu_2 = 2 \cdot\left(\frac{v}{\mu}\right) \cdots$

- $\nu =  n\left(\frac{v}{2 L}\right)$ Harmonic

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Vibrations of an Air Column in a Pipe $\rightarrow$ Vibrations of an Air Column in a Pipe $\rightarrow$ <span style = "color:blue"> Vibrations of an Air Column in a Pipe </span> $\rightarrow$ Vibrations of an Air Column in a Pipe $\rightarrow$ Vibrations of an Air Column in a Pipe </footer>

- $n=1$ First Harmonic

- $n=2$ second Harmonic

- $\left(\frac{V}{2 L}\right)=$ fundamental frequency

- $\Rightarrow \quad \nu_n=n\left(\frac{v}{2 L}\right)$

- $v=\sqrt{\frac{T}{r}}$

- $v=\gamma \sqrt{\frac{B}{\rho}}$

</div>

</div>

- $\nu =  n\left(\frac{v}{2L}\right)$ Harmonic

- $n=1$ First Harmonic

- $n=2$ second Harmonic

- $\left(\frac{V}{2 L}\right)=$ Fundamental frequency

- $\Rightarrow \quad \nu_n=n\left(\frac{v}{2 L}\right)$

- $v=\sqrt{\frac{T}{r}}$
- $v=\gamma \sqrt{\frac{B}{\rho}}$

</div>

</div>

- $y(x)=A \operatorname{Sink} \operatorname{Cos} \omega t$

- $\sin (k L) \neq 0$ it is maxismum possible displacement

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

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

</div>

</div>

- $\Delta p(x, t)=A \operatorname{Sin} k x \cos \omega t$

- $\left.k x\right|_L=k L=(2 n+1) \frac{\pi}{2} $

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

- $\nu_n=\frac{v}{\lambda}=\frac{(2 n+1)}{4 L} v=(n+\frac{1}{2})(\frac{v}{2 L})v=\gamma\sqrt{\frac{B}{\rho}}$
* B = Bulk modulus 
* $\rho$ = density

- $ \gamma=C_p / C_v$

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Vibrations of an Air Column in a Pipe $\rightarrow$ Vibrations of an Air Column in a Pipe $\rightarrow$ <span style = "color:blue"> Vibrations of an Air Column in a Pipe </span> $\rightarrow$ Physical Interpertation of the modes on a string or in an air column. $\rightarrow$ Physical Interpertation of the modes on a string or in an air column. </footer>

### <Success style="font-size:25px"> Standing Wavesina Pipe Phenomena Of Beats Doppler Effect L-4 </Success>
### <primary style="font-size:24px"> Physical Interpertation of the modes on a string or in an air column. </primary>
<hr>
<main>

- $n \frac{\lambda}{2}=L$

- $ \Rightarrow \quad \lambda=(\frac{2 L}{n})$

- <span style="color:blue">One difference between the string vibration and air column vibration.</span>

- <span style="color:green">END CORRECTION </span>
</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Vibrations of an Air Column in a Pipe $\rightarrow$ Vibrations of an Air Column in a Pipe $\rightarrow$ <span style = "color:blue"> Physical Interpertation of the modes on a string or in an air column. </span> $\rightarrow$ Physical Interpertation of the modes on a string or in an air column. $\rightarrow$ Beats </footer>

- R = radius of the pipe

- EFFECTIVE Length for ended pide = L+1.2R

- EFFECTIVE length for a pipe closed at one and = L+0.6R
</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Vibrations of an Air Column in a Pipe $\rightarrow$ Physical Interpertation of the modes on a string or in an air column. $\rightarrow$ <span style = "color:blue"> Physical Interpertation of the modes on a string or in an air column. </span> $\rightarrow$ Beats $\rightarrow$ Beats </footer>

### <Success style="font-size:25px"> Standing Wavesina Pipe Phenomena Of Beats Doppler Effect L-4 </Success>
### <primary style="font-size:24px"> Beats </primary>
<hr>
<main>

- Superposition of two waves of tHE SAME FREQUENCY
- traveling in opposite directions

- <span style="color:blue">Standing wave</span>.

- **SUPERPOSITION OF WAVES THAT HAVE SLIGHTLY DIFFERENT FREQUENCIES**

- $ \nu_1 \quad \nu_2$

- $|(\nu_1-\nu_2\)| \ll \nu_1 \sim \nu_2 $

- $ \nu_1=500 {Hz},$

- $ \nu_2=502 \{Hz} $

- <span style="color:blue">BEAT PHENOMENA</span>
</div>

![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-04-standing-waves-phenomena-of-beats-doppler-effect-l-4_5-mhskub1knc0-10.jpg)

</div>

</main>
<hr>
<footer style = "font-size: 18px">Physical Interpertation of the modes on a string or in an air column. $\rightarrow$ Physical Interpertation of the modes on a string or in an air column. $\rightarrow$ <span style = "color:blue"> Beats </span> $\rightarrow$ Beats $\rightarrow$ Beats </footer>

### <Success style="font-size:25px"> Standing Wavesina Pipe Phenomena Of Beats Doppler Effect L-4 </Success>
### <primary style="font-size:24px"> Beats </primary>
<hr>
<main>

- $y_1(x, t)=A \sin (k_1 x-\omega_1 t)$

- $y_2(x, t)=B \sin (k_2 x-\omega_2 t)$

- **We stand at a point $ x = x_0 = 0$**

- $y_1(t)=+A \sin \omega_1 t$

- $y_2(t)=B \sin \omega_2 t$

- $y(t)=y_1(t)+y_2(t)=A \sin \omega_1 t+B \sin \omega_2 t$

- $y(t)= A \sin \omega_1 t+B \sin \omega_2 t$

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Physical Interpertation of the modes on a string or in an air column. $\rightarrow$ Beats $\rightarrow$ <span style = "color:blue"> Beats </span> $\rightarrow$ Beats $\rightarrow$ Phenomena of Beats </footer>

### <Success style="font-size:25px"> Standing Wavesina Pipe Phenomena Of Beats Doppler Effect L-4 </Success>
### <primary style="font-size:24px"> Beats </primary>
<hr>
<main>

- **Beats can be understand easily if we take A =B**

- $y(t)=A[\sin \omega_1 t+\sin \omega_2 t]=2 A \sin(\frac{\omega_1+\omega_2}{2} t) \cos(\frac{\omega_1-\omega_2}{2} t)$

- $If  \omega_1=\omega_2 \quad y(t) \approx 2 A \sin \omega t $

- $\omega_1 \neq \omega_2 \quad \omega_2=\omega_1+\Delta \omega \quad \Delta \omega \ll \omega_1$

- $y(t)=2 A \sin \omega_{+} t \cdot \cos \omega_{-} t $

- $\omega_{+}=\frac{\omega_1+\omega_2}{2} \simeq \omega_1,$

- $ \omega_{-}=|(\omega_1-\omega_2)|$

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Beats $\rightarrow$ Beats $\rightarrow$ <span style = "color:blue"> Beats </span> $\rightarrow$ Phenomena of Beats $\rightarrow$ Phenomena of Beats </footer>

### <Success style="font-size:25px"> Standing Wavesina Pipe Phenomena Of Beats Doppler Effect L-4 </Success>
### <primary style="font-size:24px"> Phenomena of Beats </primary>
<hr>
<main>

- $y(t)=\quad 2 A \sin \omega_{+} t \cos \omega_{-} t$

- $\omega_{-} \ll \omega_{+}$

- $T_{+}=\frac{2 \pi}{\omega_{+}} \ll T_{-}=\frac{2 \pi}{\omega_{-}}$

- $y(t)=2 A \sin \omega_{+} t c_0 \omega_{-} t$

- <span style="color:blue"><ins>Phenomena of Beats</ins></span>
</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Beats $\rightarrow$ Beats $\rightarrow$ <span style = "color:blue"> Phenomena of Beats </span> $\rightarrow$ Phenomena of Beats $\rightarrow$ Phenomena of Beats </footer>

### <Success style="font-size:25px"> Standing Wavesina Pipe Phenomena Of Beats Doppler Effect L-4 </Success>
### <primary style="font-size:24px"> Phenomena of Beats </primary>
<hr>
<main>

- $y(x, t)=2 A \sin \omega_{+} t \cos \omega_{-} t $
 
- $\Delta p(x, t)=P \sin \omega_{+t}{\cos \omega_{-t}}$
 
 - ${(\frac{\omega_1-\omega_2}{2})}$
 
 - $\frac{T_{-}}{2}$ = $\frac{1}{2}(\frac{2 \pi}{\omega-}=\frac{2 \pi}{\omega_1-\omega_1}$
 
 - **Beat frequency $=(\omega_1-\omega_2)$**

- $\omega_1=\omega_2$   Beat frequency = 0

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Beats $\rightarrow$ Phenomena of Beats $\rightarrow$ <span style = "color:blue"> Phenomena of Beats </span> $\rightarrow$ Phenomena of Beats $\rightarrow$ Doppler Effect </footer>

### <Success style="font-size:25px"> Standing Wavesina Pipe Phenomena Of Beats Doppler Effect L-4 </Success>
### <primary style="font-size:24px"> Phenomena of Beats </primary>
<hr>
<main>

- <span style="color:blue"><ins>**Amplitude : Same for both waves**</ins></span>

- $A \sin \omega_1 t+B \sin \omega_2 t$

- $=\frac{A+B}{2} \sin \omega_1 t+\frac{A-B}{2} \sin \omega_1 t+\frac{A+B}{2} \sin \omega_2 t-\frac{(A-B )}{2} \sin \omega_2 t$ 
 
 - =$\frac{(A+B)}{2}(\sin \omega_1 t+\sin \omega_2 t)+\frac{(A-B)}{2}(\sin \omega_1 t-\sin \omega_1 t)$ 
 
 - = $\frac{(A+B)}{2} \sin \omega_{+} t \cos \omega_{-} t-\frac{(A-B)}{2} C_3 \omega_{+} t \sin \omega_{-} t$

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Phenomena of Beats $\rightarrow$ Phenomena of Beats $\rightarrow$ <span style = "color:blue"> Phenomena of Beats </span> $\rightarrow$ Doppler Effect $\rightarrow$ Doppler Effect </footer>

### <Success style="font-size:25px"> Standing Wavesina Pipe Phenomena Of Beats Doppler Effect L-4 </Success>
### <primary style="font-size:24px"> Doppler Effect </primary>
<hr>
<main>

- <span style="color:green">Source $\quad V_s$</span>

- frequency observed (heard) is different from the frequency emitted by the source

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Phenomena of Beats $\rightarrow$ Phenomena of Beats $\rightarrow$ <span style = "color:blue"> Doppler Effect </span> $\rightarrow$ Doppler Effect $\rightarrow$ Doppler Effect </footer>

### <Success style="font-size:25px"> Standing Wavesina Pipe Phenomena Of Beats Doppler Effect L-4 </Success>
### <primary style="font-size:24px"> Doppler Effect </primary>
<hr>
<main>

- **1 Source moving towards the observe**

- $\lambda=\frac{v}{2} \quad v$ speed of  wave

- $\lambda^{\prime}=\lambda-v_s T$

- $\nu_1= \frac{v}{\lambda^{\prime}}=\frac{v}{\lambda-v_s T}=\frac{v}{\frac{v}{2}-\frac{v_s}{2}}=(\frac{v}{v-v_s}) \nu$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Phenomena of Beats $\rightarrow$ Doppler Effect $\rightarrow$ <span style = "color:blue"> Doppler Effect </span> $\rightarrow$ Doppler Effect $\rightarrow$ Doppler Effect </footer>

### <Success style="font-size:25px"> Standing Wavesina Pipe Phenomena Of Beats Doppler Effect L-4 </Success>
### <primary style="font-size:24px"> Doppler Effect </primary>
<hr>
<main>

- $\nu_1=(\frac{v}{v-v_s}) \nu>\nu$

- $ \cdots-\cdot-\cdot ? $

- $\nu_1=(\frac{v}{v+v_s}) \nu<\nu$

- **3.  When an observer is moving towards the source**.

- What is the effective distance that the observer sees (feels) between the two maximum?

- Let the first maximum be emitted at time $t_1$ and the second one time $t_2$

- $t_2-t_1=T.$

- Let the observer hear the first maximum at $t_1^\prime$ and $t_2^\prime$

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Doppler Effect $\rightarrow$ Doppler Effect $\rightarrow$ <span style = "color:blue"> Doppler Effect </span> $\rightarrow$ Doppler Effect $\rightarrow$ Doppler Effect </footer>

### <Success style="font-size:25px"> Standing Wavesina Pipe Phenomena Of Beats Doppler Effect L-4 </Success>
### <primary style="font-size:24px"> Doppler Effect </primary>
<hr>
<main>

- The observer hears the second maximum at  $t_2^\prime$

- Time period falt by the observe = $t_2^\prime-t_1^\prime$

- $ t_2^{\prime}-t_1^{\prime}=\frac{\lambda}{v+v_0}$

- $T^{\prime}=\frac{1}{\nu^{\prime}}=\frac{\lambda}{(v+v_0\)}=\frac{v}{\nu(v+v_0)}$

- $\Rightarrow \quad \nu^{\prime}=(\frac{v+v_0}{v}) \nu>\nu$

- $\nu^{\prime}=(\frac{v+v_0}{v}) \nu = (\frac{v-v_0}{v}) \nu<\nu$

- $2^{\prime}=(\frac{v_{ \pm} v_0}{v \pm v_s}) 2$

</div>

</div>