Waves (11Acd13)

Sheet 03 Displacement Relationship, Particle Velocity and Acceleration, Phase and Phase Difference, Speed Of Transverse Wave

1. The equation of a plane progressive wave is \(y=10Sin2\pi\left(t-0.005x\right)\), where \(y\) and \(x\) are in \(cm\) and \(t\) is in \(seconds\). calculate: the (i) Amplitude. (ii) frequency. (iii) Wavelength and (iv) Velocity of wave.
   
   Ans: (i) \(r=10cm\)
   (ii) \(\nu=1 Hz\)
   (iii) \(\lambda=200cm\) and
   (iv) \(V=200\frac{cm}{s}\).
   
   
2. A wave is travelling along a string is described by the equation \(y\left(x,t\right)=0.005Sin\left(80.0x-3.0t\right)\) in which the numerical constants are in S.I units \(0.005m,80\frac{rad}{m}\) and \(3.0\frac{rad}{s}\). Calculate: (a) Amplitude. (b) Wavelength. (c) Frequency. (d) time Period. (e) Wave Velocity and (f) Amplitude of the particle velocity. Also, Calculate: (g) the displacement of the wave at a distance \(x=30.0cm\) and time \(t=20s\).
   
   Ans: (a) \(r=0.005m\)
   (b) \(\lambda=7.85cm\)
   (c) \(T=2.09Sec\)
   (d) \(\nu=0.48Hz\)
   (e) \(v=\nu\lambda\)
   (f) \(V_{amplitude}=0.015\frac{m}{s}\) and
   (g) \(\left(-0.005Sin36\right)m\).
   
   
3. A harmonically moving transverse wave on a string has maximum particle velocity of \(3\frac{m}{s}\) and maximum particle acceleration is \(90\frac{m}{s^{2}}\). Velocity of the wave is \(20\frac{m}{s}\). What is the equation of the wave form?
   
   Ans: \(y=0.1Sin\left(30t+1.5x\right)\).
   
   
4. For a travelling harmonic wave \(y=6cos\left(5t-0.09x+0.4\right)\). Where; \(x\) and \(y\) are in \(cm\) and \(t\) is in \(Seconds\). What is the phase difference between oscillatory motion at two points separated by a distance of (i) \(2m\) (ii) \(0.2m\) (iii) \(\frac{\pi}{3}\) and (iv) \(\frac{\pi}{4}\)?
   
   Ans: (i) \(17.95rad\)
   (ii) \(1.79rad\)
   (iii) \(\frac{2\pi}{3}rad\) and
   (iv) \(\frac{\pi}{2}rad\).
   
   
5. Write the equation of progressive wave propagating along x-direction, whose amplitude is \(5cm\), frequency \(250Hz\) and Velocity \(500\frac{m}{s}\)?
   
   Ans: \(y=0.05Sin\pi\left(500t-x\right)m\).
   
   
6. For a travelling harmonic wave, \(y=2.0Cos\left(10t-0.0080x+0.18\right)\) where \(x\) and \(y\) are in \(cm\) and \(t\) is in \(Seconds\). What is the phase difference between two points separated by (i) A distance of \(0.5m\) and (ii) a time gap of \(0.5Sec\)?
   
   And: (i) \(\Delta\Phi_{0.5m}=-0.4rad\) and (ii) \(\Delta\Phi_{0.5s}=5rad\).
   
   
7. What will be the displacement of an air particle \(3.5m\) from the origin of the disturbance at \(t=0.05s\), when a wave of amplitude \(0.2m\) and frequency \(500Hz\) travels along it with a velocity \(350\frac{m}{s}\)?
   
   Ans: \(y=o(Zero)\).
   
   
8. A simple harmonic wave train of amplitude \(2cm\) and frequency \(100Hz\), is travelling in positive X-direction with a velocity of \(15\frac{m}{s}\). What will be (a) The displacement (b) Particle Velocity at \(x=180cm\) from the origin at \(t=5s\).
   
   Ans: (a) \(y=0(Zero)\) (b) \(v=400\pi\frac{cm}{s}\).
   
   
9. For a plane wave \(y=2.5\times10^{-0.02x}Cos\left(800t-0.82x+\frac{\pi}{2}\right)\). Write Down (a) The general expression for phase \(\Phi\). (b) The phase at \(x=0\) and \(t=0\). (c) The phase difference between the points separated by \(20cm\) along X-axis. (d) The change in phase at a \(\Delta t=0.6ms\). (e) The amplitude at \(x=100m\).
   Take: Units of \(y,t\) and \(x\) as \(10^{-5}cm\), \(Seconds\) and \(metre\) respectively.
   
   
   Ans: (a) \(\Phi=\left(800t-0.82x+\frac{\pi}{2}\right)\)
   (b) \(\Phi_{x=0,t=0}=\frac{\pi}{s}rad\)
   (c) \(\Delta\Phi_{(\Delta x=20cm)}=-0.164rad\)
   (d) \(\Delta\Phi_{(\Delta t=0.6ms)}=0.48rad\) and
   (e) \(r=\left(0.025\times10^{-5}\right)cm\).