Question

. The equation of a stationary wave on a string fixed at both ends is given by
y( x,t) = 2sin πx cos100πt
where x and y are measured in metre and t in second. Calculate the amplitude,
wavelength and frequency of component waves whose superposition generated this
stationary wave. Also write the equations of component waves.

Answer 1

Question # 85455, Physics / Mechanics | Relativity

Task: The equation of a stationary wave on a string fixed at both ends is given by $y(x,t)=2\sin(\pi x)\cos(100\pi t)$, where $x$ and $y$ are measured in metre and $t$ in second. Calculate the amplitude, wavelength and frequency of component waves whose superposition generated this stationary wave. Also write the equations of component waves.

Solution:

$y(x,t)=2\sin(\pi x)\cos(100\pi t)=\sin(\pi x+100\pi t)+\sin(\pi x-100\pi t)=\sin(\pi x+100\pi t)+\sin(100\pi t-\pi x+\pi)$

General formula for a standing wave: $y(x,t)=A\sin(\omega t-kx+\phi)$.

We have $y(x,t)=y_{1}(x,t)+y_{2}(x,t)$.

Consider first component $y_{1}(x,t)=\sin(\pi x+100\pi t)$:

amplitude is $A_{1}=1$ m

angular frequency is $\omega_{1}=100\pi$ rad/s

frequency is $f_{1}=\omega_{1}/2\pi=50$ Hz

wave number is $k_{1}=-\pi$ rad/m

velocity is $v_{1}=\omega_{1}/k_{1}=-100$ m/s

(velocity is negative because the wave is travelling in the negative $x$ direction)

wavelength is $\lambda_{1}=|v_{1}|/f_{1}=2$ m

Consider second component $y_{2}(x,t)=\sin(100\pi t-\pi x+\pi)$:

amplitude is $A_{2}=1$ m

angular frequency is $\omega_{2}=100\pi$ rad/s

frequency is $f_{2}=\omega_{2}/2\pi=50$ Hz

wave number is $k_{2}=\pi$ rad/m

velocity is $v_{2}=\omega_{2}/k_{2}=100$ m/s

wavelength is $\lambda_{2}=|v_{2}|/f_{2}=2$ m