Question

soln plzzzzzzzzzzzzzzz@zzzz​

Answer 1

Plane Progressive Waves

A plane progressive wave is one that moves in space, but lies in a single plane all the time. It has a constant amplitude and a single direction of propagation.

The Particle Displacement y for a plane progressive wave can be represented by the following equation:

$y=A\sin(\omega t-kx)$

where

$A = \textsf{Amplitude} \\ \\ \omega = \textsf{Angular Frequency} = 2\pi f \\\\ k = \textsf{Angular Wave Number} = \frac{2\pi}{\lambda}$

Taking the $2\pi$ factor common, we have:

$\begin{aligned}\rightarrow y &= A\sin(\omega t-kx) \\\\\\ &= A\sin \left(2\pi ft - \dfrac{2\pi}{\lambda}x\right) \\\\\\ \therefore y &= A\sin 2\pi \left(ft-\frac{1}{\lambda}x\right)\end{aligned}$

Now, Let's take a look at our given equation:

$y=10\sin 2\pi (t-0.005x) \\\\\\ \textsf{Comparing with standard form } {\sf y=A\sin 2\pi\left(ft-\dfrac{1}{\lambda}x\right)} \\\\\\\Large \rightarrow \boxed{A = 10 \: cm} \\\\\\ \rightarrow \boxed{f = 1\ s^{-1} = 1\ Hz} \\\\\\ \rightarrow \dfrac{1}{\lambda}=0.005\ cm^{-1} \\\\\\ \implies \lambda = \dfrac{1}{0.005} = 200\ cm = 2\ m \\\\\\ \implies \boxed{\lambda=2\ m}$

Also, Wave Velocity, say $v$, is given by:

$v = \lambda f \\\\\\ \implies v = 2\times 1\ m/s \\\\\\ \implies \Large \boxed{v=2\ m/s}$

Thus, we found everything we needed. Here's the summary of final results:

$\large\underline{\texttt{SUMMARY OF RESULTS}}\\\boxed{\begin{minipage}{20em}\tt Amplitude = A = 10 cm \\\\\\ Frequency = f = 1 Hz \\\\\\ Wavelength = \lambda = 200 cm = 2 m\\\\\\ Wave Velocity = v = 200 cm/s = 2 m/s \end{minipage}}$