16.2 Mathematics of waves  (Page 6/11)

Solution

1. Write the wave function of the second wave: $y_2\left(x,t\right)=A\text{sin}\left(2kx+2\omega t\right).$
2. Write the resulting wave function:
   
   $y_R\left(x,t\right)=y_1\left(x,t\right)+y\left(x,t\right)=A\text{sin}\left(kx-\omega t\right)+A\text{sin}\left(2kx+2\omega t\right).$
3. Find the partial derivatives:
   
   $\begin{array}{ccc}\hfill \frac{\partial y_R\left(x,t\right)}{\partial x}& =\hfill & \text{−}Ak\text{cos}\left(kx-\omega t\right)+2Ak\text{cos}\left(2kx+2\omega t\right),\hfill \\ \hfill \frac{{\partial }^2{y}_R\left(x,t\right)}{{\partial }^{2}x}& =\hfill & \text{−}A{k}^2\text{sin}\left(kx-\omega t\right)-4A{k}^2\text{sin}\left(2kx+2\omega t\right),\hfill \\ \hfill \frac{\partial y_R\left(x,t\right)}{\partial t}& =\hfill & \text{−}A\omega \text{cos}\left(kx-\omega t\right)+2A\omega \text{cos}\left(2kx+2\omega t\right),\hfill \\ \hfill \frac{{\partial }^2{y}_R\left(x,t\right)}{{\partial }^{2}t}& =\hfill & \text{−}A{\omega }^2\text{sin}\left(kx-\omega t\right)-4A{\omega }^2\text{sin}\left(2kx+2\omega t\right).\hfill \end{array}$
4. Use the wave equation to find the velocity of the resulting wave:
   

   $\begin{array}{ccc}\hfill \frac{{\partial }^{2}y\left(x,t\right)}{\partial x^2}& =\hfill & \frac{1}{v^2}\frac{{\partial }^{2}y\left(x,t\right)}{\partial t^2},\hfill \\ \hfill -A{k}^2\text{sin}\left(kx-\omega t\right)-4A{k}^2\text{sin}\left(2kx+2\omega t\right)& =\hfill & \frac{1}{v^2}\left(\text{−}A{\omega }^2\text{sin}\left(kx-\omega t\right)-4A{\omega }^2\text{sin}\left(2kx+2\omega t\right)\right),\hfill \\ \hfill k^2\left(\text{−}A\text{sin}\left(kx-\omega t\right)-4A\text{sin}\left(2kx+2\omega t\right)\right)& =\hfill & \frac{{\omega }^2}{v^2}\left(\text{−}A\text{sin}\left(kx-\omega t\right)-4A\text{sin}\left(2kx+2\omega t\right)\right),\hfill \\ \hfill k^2& =\hfill & \frac{{\omega }^2}{v^2},\left|v\right|=\frac{\omega }{k}.\hfill \end{array}$

Significance

Any disturbance that complies with the wave equation can propagate as a wave moving along the x -axis with a wave speed v . It works equally well for waves on a string, sound waves, and electromagnetic waves. This equation is extremely useful. For example, it can be used to show that electromagnetic waves move at the speed of light.

Summary

• A wave is an oscillation (of a physical quantity) that travels through a medium, accompanied by a transfer of energy. Energy transfers from one point to another in the direction of the wave motion. The particles of the medium oscillate up and down, back and forth, or both up and down and back and forth, around an equilibrium position.
• A snapshot of a sinusoidal wave at time $t=0.00\text{s}$ can be modeled as a function of position. Two examples of such functions are $y\left(x\right)=A\text{sin}\left(kx+\varphi \right)$ and $y\left(x\right)=A\text{cos}\left(kx+\varphi \right).$
• Given a function of a wave that is a snapshot of the wave, and is only a function of the position x , the motion of the pulse or wave moving at a constant velocity can be modeled with the function, replacing x with $x\mp vt$ . The minus sign is for motion in the positive direction and the plus sign for the negative direction.
• The wave function is given by $y\left(x,t\right)=A\text{sin}\left(kx-\omega t+\varphi \right)$ where $k=2\pi \text{/}\lambda$ is defined as the wave number, $\omega =2\pi \text{/}T$ is the angular frequency, and $\varphi$ is the phase shift.
• The wave moves with a constant velocity $v_w$ , where the particles of the medium oscillate about an equilibrium position. The constant velocity of a wave can be found by $v=\frac{\lambda }{T}=\frac{\omega }{k}.$

Conceptual questions