| 495 |
wave function (general form).
the positive sign means the wave is
traveling in the negative $x$ direction,
the negative sign in the positive $x$ direction.
|
\[
y=y\left(x,t\right)=f\left(x\pm v t\right)
\]
|
| 495 |
equations of motion of an initial wave point
(say the crest) of a transversal wave at a later
time $t,$ if the point was initially located at
coordinate $x_0$ at time $t=0.$ Here we are
assuming that $y$ is defined by
$y(x,t)\equiv f(x\pm vt).$
|
\[
x=x_0\pm vt,\ \ \left(+\mathrm{\ right\ motion,}-\mathrm{left}\right)\\
x_0=x\mp vt\\
y\left(x_0\pm v t,t\right)=f\left(\left[x_0\mp v t\right]\pm v t\right)=f\left(x_0\right)
\]
|
| 497 |
superposition of waves
|
\[
y\left(x_0,t_0\right)=y_1\left(x_0,t_0\right)+y_2\left(x_0,t_0\right)
\]
|
| 504 |
sinusoidal wave function for a
mechanical transversal wave
(vertical displacement)
with
$y = 0$ when $x = 0$ and $t = 0.$
$A=$ amplitude,
$\lambda=$ wavelength,
$k=$ angular wave number,
$\omega=$ angular frequency,
$T=$ period
|
\[
\begin{align*}
y &=A\sin\left[\frac{2\pi}{\lambda}\left(x-vt\right)\right]\\
&=A\sin\left[2\pi\left(\frac{x}{\lambda} - \frac{t}{T}\right)\right]\\
&=A\sin\left(kx-\omega t\right)
\end{align*}
\]
|
| 504 |
sinusoidal wave function for a mechanical
transversal wave (vertical displacement)
with phase constant $\phi.$
$A=$ amplitude,
$\lambda=$ wavelength,
$k=$ angular wave number,
$\omega=$ angular frequency,
$T=$ period
|
\[
y=A\sin{\left(kx-\omega t+\phi\right)}
\]
|
| 500 |
speed of a wave on a taut string.
$\mu=$ mass per unit length,
$T=$ tension
in the string (not period!)
|
\[
v=\sqrt{\frac{T}{\mu}}
\]
|
504
496 |
speed of wave.
|
\[
v=\frac{dx}{dt}=\frac{\omega}{k}=\frac{\lambda}{T}=\lambda f
\]
|
| 504 |
angular number.
|
\[
k\equiv\frac{2\pi}{\lambda}
\]
|
| 504 |
angular frequency.
|
\[
\omega\equiv\frac{2\pi}{T}=2\pi f
\]
|
| 506 |
displacement speed of transverse wave (simple harmonic motion)
|
\[
v_y=\frac{\partial y}{\partial t}=-\omega A\cos{\left(kx-\omega t\right)}
\]
|
| 506 |
displacement acceleration of transverse wave (simple harmonic motion)
|
\[
a_y=\frac{\partial^2y}{\partial t^2}=\frac{\partial v_y}{\partial t}=-\omega^2A\sin{\left(kx-\omega t\right)}
\]
|
| 506 |
maximum displacement speed (simple harmonic motion).
Occurs when $y=0.$
|
\[
v_{y,\mathrm{max}}=\omega A
\]
|
| 506 |
maximum displacement acceleration (simple harmonic motion).
Occurs when $y=\pm A.$
|
\[
a_{y,\mathrm{max}}=\omega^2A
\]
|
| 508 |
total potential energy in one wavelength of a string wave
|
\[
U_\lambda=\frac{1}{4}\mu\omega^2A^2\lambda
\]
|
| 508 |
total kinetic energy in one wavelength of a string wave
|
\[
K_\lambda=\frac{1}{4}\mu\omega^2A^2\lambda
\]
|
| 508 |
total energy in one wavelength.
This amount of energy passes
by a given point on the string during one period of oscillation.
|
\[
E_\lambda=U_\lambda+K_\lambda=\frac{1}{2}\mu\omega^2A^2\lambda
\]
|
| 508 |
power transmitted by a sinusoidal wave on a stretched string
|
\[
\mathscr{P}=\frac{1}{2}\mu\omega^2A^2v
\]
|
| 510 |
linear wave equation for a wave on a string.
$T=$ tension
|
\[
\frac{\mu}{T}\frac{\partial^2y}{\partial t^2}=\frac{\partial^2y}{\partial x^2}
\]
|
| 510 |
linear wave equation in general
|
\[
\frac{\partial^2y}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2y}{\partial t^2}
\]
|
Alex Notes