Question

the transverse displacement (y) of a wave is given as a function of position (x in meters) and time (t in seconds) by the expression. determine the wavelength, frequency, period, and phase constant of this waveform.

$y(x,t)=y_{in}\sin(0.197x + 801t + 4.37)$

Explanation:

Step1: Identify wave number \(k\) and angular frequency \(\omega\)

The general form of a wave is \(y(x,t)=y_{m}\sin(kx+\omega t+\phi_{0})\). Comparing with \(y(x,t) = y_{m}\sin(0.197x + 801t+4.37)\), we have \(k = 0.197\space m^{-1}\) and \(\omega=801\space rad/s\)

Step2: Calculate wavelength \(\lambda\)

The formula for wavelength is \(\lambda=\frac{2\pi}{k}\). Substituting \(k = 0.197\space m^{-1}\), we get \(\lambda=\frac{2\pi}{0.197}\approx31.9\space m\)

Step3: Calculate frequency \(f\)

The formula for frequency is \(f=\frac{\omega}{2\pi}\). Substituting \(\omega = 801\space rad/s\), we get \(f=\frac{801}{2\pi}\approx127.5\space Hz\)

Step4: Calculate period \(T\)

The formula for period is \(T=\frac{1}{f}\). Since \(f=\frac{\omega}{2\pi}\), then \(T=\frac{2\pi}{\omega}\). Substituting \(\omega = 801\space rad/s\), we get \(T=\frac{2\pi}{801}\approx0.00785\space s\)

Step5: Identify phase constant \(\phi_{0}\)

From the given equation \(y(x,t) = y_{m}\sin(0.197x + 801t+4.37)\), the phase constant \(\phi_{0}=4.37\space rad\)

Answer:

\(\lambda = 31.9\) meters, \(f = 127.5\) Hertz, \(T = 0.00785\) seconds, \(\phi_{0}=4.37\) radians