physics-class-11-unit-15-chapter-02-intro-to-waves-equation-sinusodial-and-speed-of-waves-l-2-5-yfc57wjpjdc

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Waves </primary>
<hr>
<main>

- <span style="color:green">What is a Wave?</span>

- <span style="color:blue">Is it particles moving from one place to another?</span>

- When a person is speaking to another person, sound travels but the air in better dos not.

- <span style="color:green">A wave does not represent movement of particles.</span>
</div>

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px"> $\rightarrow$ Introduction to Waves Equation, Sinusodial and Speed of Waves $\rightarrow$ <span style = "color:blue"> Waves </span> $\rightarrow$ Waves $\rightarrow$ Waves </footer>

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Waves </primary>
<hr>
<main>

- <span style="color:blue">A wave is a disturbance that travels from one place to another!</span>

- <span style="color:green">Dose a wave mean that the distarbance has to travel from one place to another?</span>

- wave is a distarbance created at one place and travelling to onther place (<span style="color:green">travelling waves</span>)

- <span style="color:blue">An extanded distarbance</span> (<span style="color:green">stationary waves</span>)

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Introduction to Waves Equation, Sinusodial and Speed of Waves $\rightarrow$ Waves $\rightarrow$ <span style = "color:blue"> Waves </span> $\rightarrow$ Waves $\rightarrow$ Pulse </footer>

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Pulse </primary>
<hr>
<main>

- $y(x) = a^2- x^2 \quad |x|\geq a $

- $\quad$ = 0 otherwise

- $ y(x, t)=a^2-(x-x_0)^2 \quad|x-x_0| \leqslant a $
 
- $\quad$ = 0
 
- $ a^2-(x-v t)^2  \mid x-vt\mid \leqslant$
 
- $\quad $ = 0 otherwise

- $y(x, t) =  a^2-(x+v t)^2 \quad |x+v t|\leqslant a $

- $\quad $ =  0  otherwise

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Waves $\rightarrow$ Pulse $\rightarrow$ <span style = "color:blue"> Pulse </span> $\rightarrow$ Pulse $\rightarrow$ Wave Travel </footer>

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Pulse </primary>
<hr>
<main>

- $y(x, t)$ is a function of either $(x-u t) or(x+vt)$

- $f(x, t)=f(x+v t, t=0)$

- A disturbance created as $f(x)$ at time $t=0$ will be changing / will be give a a function of time as

- $f(x, t)=f(x-v t, t=0)$

- $y$ it travels $t$ the right $f(x, t)$

- $=f(x+v t, t=0)$ if it travels $t$ the left.

</div>

</div>

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

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Wave Travel </primary>
<hr>
<main>

* <span style="color:green"> $f(x-v t, t=0)=f(x, t)$</span>
* <span style="color:blue"> $f(x+v t, 0)=f(x t)$</span>
 
- **<ins>Condition:</ins>**

- <span style="color:green">The disturbance travels undistorted with constant speed $v$. </span>

- <span style="color:blue">Undistorted $\Longleftrightarrow $ Dispersionless</span>

- <span style="color:blue">Another way of looking at wave travel.</span>

- <span style="color:blue">$\Delta t=\frac{x}{v}$</span>

- $f(x, t)=f\left(x=0, t-\frac{x}{v}\right)$  If travel is to the right.

</div>

</div>

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

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Wave Travel </primary>
<hr>
<main>

- $f(x, 1)=f(x=0, t + \frac{x}{v}) \quad x<0$

- <span style="color:green">wave as a function</span>

- $f(t+\frac{x}{v})$, wave disturbance travelling to the left.

- $f(t-\frac{x}{v})$, wave disturbance travelling to the right.

- $f(t+\frac{x}{v})=f(\frac{v t+x}{v})=f(x+v t) $

- $f(t-\frac{x}{v})=f(\frac{v t-x}{v})=f(v t-x)$

- <span style="color:green">A wave can be described as</span> $f(x+v t), f(x-v t)$

- $f(t+x / v)$, or $(t-\frac{x}{v})$.

</div>

</div>

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

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Wave Travel </primary>
<hr>
<main>

- <span style="color:blue">$y(x, t) = A \sin (2 \pi(\frac{x}{\lambda}-f t)$</span>

- $f(x-v t)$ are functions of $x$, and $t$ both.

- <span style="color: blue">A travelling is $a$ function of $x$ ane $t$</span>

- Look at $f(x-v t)$ as a function of time by looking at $a$ fixed position $x_0$

- See how wave disturbance to changing with time.
</div>

</div>

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

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Wave </primary>
<hr>
<main>

- $f(x=0, t)=  A \sin (\frac{0}{\lambda}-f t)$

- $ =-A \sin 2 \pi f t $

- $ =-A \sin \omega t $

- As a function of position: $t=0 $.

- $ f(x, t=0)  =A \sin \frac{2 \pi}{\lambda} x $

- $ x \rightarrow (x+\lambda) \quad A \sin \frac{2 \pi}{\lambda}(x+\lambda)$

- $ =A \sin \left(\frac{2 \pi}{\lambda} x+2 \pi\right) $

- $ =A \sin \frac{2 \pi}{\lambda} x$

</div>

</div>

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

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Wave </primary>
<hr>
<main>

- <span style="color:green">How are $\lambda, v$ and frequency $f$ of the ware related </span>
 
- $x=x_0$

- $A \sin \omega t$

- After time $t=T=\frac{1}{f}$

- The displacement is the same.

- In time $T$, the ware has moves by distance $\lambda$

- $v=\frac{\lambda}{T}=f \lambda$

</div>

</div>

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

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Waves </primary>
<hr>
<main>

- How to generate $a$ wave like $A \sin 2 \pi\left(\frac{x}{\lambda}-f t\right)$.
- This is called a sine wave.

- $y(x=0, t)  =A \sin 2 \pi f t $

- $y(x, t)  =y(x=0, t-\frac{x}{v})$

- $ =A \sin 2 \pi f(t-\frac{x}{v}) $

- $ =A \sin 2 \pi(f t-\frac{x f}{v}) $

- $ =-A \sin 2 \pi(\frac{x}{\lambda}-f t)$

</div>

</div>

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

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Sinusoidal Wave </primary>
<hr>
<main>

- $ y(x, t)= y(x=0, t+\frac{x}{v})$

- $ ={A} \sin 2 \pi+(t+\frac{x}{v}) $

- $ =A \sin 2 \pi(f t+\frac{\pi}{\lambda})$.

- (1) We have looked at a disturbance travelling with speed $v$ are undisturbed.

- $f(x-v t) or f(t-\frac{x}{v})$

- If it travels to the right (+x axis)

- $f(x+v t) \text { or } f(t+\frac{x}{v})$

- y travelling to the left.
</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Wave $\rightarrow$ Waves $\rightarrow$ <span style = "color:blue"> Sinusoidal Wave </span> $\rightarrow$ Sinusoidal Wave $\rightarrow$ Types of Waves </footer>

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Sinusoidal Wave </primary>
<hr>
<main>

- $y(x, t)=A \sin 2 \pi\left(\frac{x}{\lambda}-f t\right)$

- $ V=f \lambda=(2 \pi f) \frac{\lambda}{2 \pi}$

- $ \frac{2 \pi}{\lambda}= k $ = wave vector 
 
- wave number = $\frac{\omega}{k}$

- $ \omega=v k $
 
- $ y(x, t)=A \sin(k x-\omega t)$
</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Waves $\rightarrow$ Sinusoidal Wave $\rightarrow$ <span style = "color:blue"> Sinusoidal Wave </span> $\rightarrow$ Types of Waves $\rightarrow$ Speed of Wave </footer>

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Types of Waves </primary>
<hr>
<main>

* <ins>Transverse waves</ins>: 
 
- $y(x,t)$ is perpendicular to the travel direction wave on a string are example of transverse waves.

* <ins>Longitudinal waves:</ins>
 
-  Distarbance is in the same direction as the direction of motion of wave.

- Sound waves are longitudinal.

- Question : Do waves Carry particles as they travel?

- Question : Do waves carry energy from one place to another?

</div>

</main>
<hr>
<footer style = "font-size: 18px">Sinusoidal Wave $\rightarrow$ Sinusoidal Wave $\rightarrow$ <span style = "color:blue"> Types of Waves </span> $\rightarrow$ Speed of Wave $\rightarrow$ Speed of Wave </footer>

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Speed of Wave </primary>
<hr>
<main>

- Speed of waves on a string.

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

- $ v=\frac{\omega}{k} \text { or } \omega=v k$

- (Assume $y \ll \lambda)$ $\theta \ll 1 $

- $\Rightarrow \sin \theta=\tan \theta =(\frac{d y}{d x})$

- $\cos \theta  =1$

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Sinusoidal Wave $\rightarrow$ Types of Waves $\rightarrow$ <span style = "color:blue"> Speed of Wave </span> $\rightarrow$ Speed of Wave $\rightarrow$ Wave </footer>

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Speed of Wave </primary>
<hr>
<main>

- Net vertical force:

- ${[T(\frac{d y}{d x})_2-T(\frac{d y}{d x})_2] \hat{y}}$

- = 0 in  the x direction

- Taylor's theorm $\(\frac{d y}{d x})_2 = (\frac{d y}{d x})_1 + (\frac{d^2 y}{d x}) \Delta x$

- Vertical force on the section of string

- $=T(\frac{d^2 y}{d x^2}) \Delta x$

- Acceleation of the section = $\frac{d^2y}{dt^2}$
</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Types of Waves $\rightarrow$ Speed of Wave $\rightarrow$ <span style = "color:blue"> Speed of Wave </span> $\rightarrow$ Wave $\rightarrow$ Wave </footer>

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Wave </primary>
<hr>
<main>

- Force $=T(\frac{d^2 y}{d x^2})_t=T(\frac{\partial^2 y}{\partial x^2})_t {\Delta x}$

- Acceleration : $(\frac{d^2 y}{d t^2})_x =(\frac{\partial^2 y}{\partial t^2})_x$

- Force $=\mu \Delta x$ (accelaration)

- $\mu=$ mass per unit length of string

- $ T(\frac{d^2 y}{d x^2})_t {\Delta x}=\mu \Delta x(\frac{d^2 y}{d t^2})_x $

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

- $T(\frac{d^2 y}{d x})_t \Delta x=\mu \Delta x^2(\frac{d^2 y}{d t^2})_x$

</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-02-intro-to-waves-equation-sinusodial-and-speed-of-waves-l-2_5-yfc57wjpjdc-21.jpg)
![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-02-intro-to-waves-equation-sinusodial-and-speed-of-waves-l-2_5-yfc57wjpjdc-22.jpg)

</div>

</main>
<hr>
<footer style = "font-size: 18px">Speed of Wave $\rightarrow$ Speed of Wave $\rightarrow$ <span style = "color:blue"> Wave </span> $\rightarrow$ Wave $\rightarrow$ Sound waves </footer>

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Wave </primary>
<hr>
<main>

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

- $(\frac{d^2 y}{d x^2})_t=-k^2 A \sin (k x-\omega t)$

- $(\frac{d^2 y}{d t^2})_x=-\omega^2 A \sin (k x-\omega t)$

- $T k^2= \mu \omega^2-(\frac{\omega}{k})=\sqrt{\frac{T}{\mu}}$

- $\Rightarrow \quad v=\frac{\omega}{k}=\sqrt{\frac{T}{\mu}}$

- $\Rightarrow \lambda=\frac{v}{f}=\frac{1}{f}\sqrt{\frac{T}{\mu}}$

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Speed of Wave $\rightarrow$ Wave $\rightarrow$ <span style = "color:blue"> Wave </span> $\rightarrow$ Sound waves $\rightarrow$ Sound waves </footer>

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Sound waves </primary>
<hr>
<main>

- m = $\rho A \Delta x$, acceleration ( $\frac{\partial^2 f}{\partial t^2})$

- $(\Delta P A)=$ Force

- $A\rho \Delta x\frac{\partial^2 f}{\partial t^2}=-\Delta Pf A$

- $\rho \frac{\partial^2 f}{\partial \omega^2} =-(\frac{\Delta P}{\Delta x})$

- Pressure ' $P$ ' change the volume

- $B=-V(\frac{\partial P}{\partial V}) =-V(\frac{\Delta P}{\Delta V})$

</div>

</div>

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

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Sound waves </primary>
<hr>
<main>

- $\overline{\Delta p}  =p $

- $ \Delta V  =A(z+\Delta z)-A z $

- $ =(A \cdot \Delta z) $

- $= A(\frac{d z}{d x}) \Delta x$

- $ -V(\frac{P}{\Delta V})  = B=-A \Delta x \cdot \frac{p}{A(\frac{d z}{d x}) \Delta x}$

- $\Rightarrow P = - B \frac{dz}{dx}$

- $\rho \frac{\partial^2 f}{\partial t^2}=-(\frac{\Delta p}{\Delta x})=-(\frac{d p}{d x})$

- $ P=-B(\frac{d z}{d x}) $

</div>

</div>

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

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Sound waves </primary>
<hr>
<main>

- $\rho \frac{\partial^2 z}{\partial t^2}=-(\frac{d p}{d x})_t=+B(\frac{d^2 z}{d x^2})$

- $(\frac{d^2 z}{d x^2})=(\frac{\rho}{B})(\frac{d^2 z}{d t^2})$

- <span style="color:blue">$\frac{d^2 z}{d t^2}=(\frac{B}{\rho})(\frac{d^2 z}{d x^2}) \quad p \propto z$</span>

</div>

</div>

### <Success style="font-size:25px"> Intro To Waves Equation Sinusodial And Speed Of Waves L-2 </Success>
### <primary style="font-size:24px"> Sound waves </primary>
<hr>
<main>

- Taking Sinusoidal wave

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

- $ \frac{d^2 z}{d x^2}|_{-2}=-k^2 A \sin (k x-\omega t) $

- $ \frac{d^2 z}{d t^2}|_x=-\omega^2 A \sin (k x-\omega t)$

- <span style="color:blue">$ \frac{d^2 f}{d t^2}=\frac{d^2 z}{d t^2}=\frac{B}{\rho} \frac{d^2 z}{d x^2}$</span>

- <span style="color:blue">$ -\omega^2=\frac{\beta}{\rho} \cdot k^2 $</span>

- <span style="color:blue">$ v^2=(\frac{\omega^2}{k^2})$</span><span style="color:blue">$=B / \rho-\sqrt{v}=\sqrt{\frac{B}{\rho}}$</span>

- B = bulk modulus

- <span style="color:blue">(Adiabatic bulk modulus)</span>

- <span style="color:blue">$v=\sqrt{\frac{B}{\rho}}$</span>

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Sound waves $\rightarrow$ Sound waves $\rightarrow$ <span style = "color:blue"> Sound waves </span> $\rightarrow$ Conclude $\rightarrow$ Thank You </footer>