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Mechanical Waves

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Attention Physics Dabblers!

Please accept the challenge of the following question:

"When a body of water sustains waves, why do the
water particles move in circles?"

Send us a brief description of a correct model for this process.  Thank You!


Previously Asked Questions

Q:     "When using the equation for the displacement of a particle in a progressive wave :
x = A cos ( w t + f ),where f is the phase angle, what exactly is the phase angle, and how do you use it in calculations? "

A:     The "Phase Angle" or "Phase Shift" f is expressed in radians, like the main angle wt. At any given moment in time, t, wt has a certain given value, and you have to add to it the value of the phase shift, f. After you add them, you calculate cosine of the total angle and plug it into the equation of the wave. The meaning of this "Phase Shift" f becomes clear if you consider the initial moment in time (t = 0) when you start observing the behavior of the wave. For t = 0, wt is equal to zero, and you have to calculate only cos (0 + f )=cos ( f ), which, after substituting into the wave equation gives you x = A cos ( f ) as the value of the position x at the initial moment in time (t=0). If this "Phase Shift" f is zero, then x at t=0 is x=0. In general, at t=0, x doesn't have to be equal to zero; its value is A cos ( f ).

Q:     How do transverse waves differ from longitudinal waves?

A:     In transverse waves, the components of the medium oscillate in a direction perpendicular to the direction of propagation of the wave through the medium.  Example: The waves in stretched strings.  In longitudinal waves, the components of the medium oscillate in a direction parallel to the direction of propagation of the wave through the medium.   Example: Sound waves in columns of air.

Q:    In an earthquake both transverse and longitudinal waves are sent out.   The transverse waves travel through the earth more slowly than the longitudinal waves (5 km/s and 9 km/s respectively.)  By detecting the time of arrival of the waves, how can the distance to the epicenter be determined?  How many detection centers are necessary to pinpoint the location of the epicenter?

A:  The two waves start at the same point and travel some distance (d) to a detection center.  Since the longitudinal wave travels faster than the transverse wave it will arrive at the detection center first.  The detection center will then begin recording the time from when the longitudinal wave hits to when the transverse wave hits.  By using the equation: d = 1.13 km/s delta.gif (839 bytes)t  (derived below, the difference in time (delta.gif (839 bytes)t) will give the distance (d) to the epicenter.

 equ1.gif (2891 bytes)
epiA.gif (48283 bytes)

This equation (d = 1.13 km/s delta.gif (839 bytes)t) gives the distance to the epicenter.   It does not however tell in which direction the quake arrived.  The quake could have hit anywhere along a circle at distance d from the receiving station.   To pinpoint the quake's epicenter 3 detection centers are needed.  Each detection center will have its own distance from the epicenter and thus its own circle where the quake could have occurred.  Where the three circles intersect is the epicenter of the earthquake.

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References

Equations

Sinusoidal waves

y(x,t) = ym sin(kx - omega2.gif (834 bytes)t)

ym is the amplitude omega2.gif (834 bytes) is the angular frequency
k is the angular wave number (kx - omega2.gif (834 bytes)t) is the phase
Wavelength (lambda2.gif (834 bytes)) related to k k = 2pi2.gif (831 bytes) / lambda2.gif (834 bytes)
Period and Angular Frequency related to omega2.gif (834 bytes) omega2.gif (834 bytes) / 2pi2.gif (831 bytes) = f = 1/T
Wave Speed v = omega2.gif (834 bytes) / k
Equation of a traveling wave y(x,t) = h (kx plusminus.gif (52 bytes) omega2.gif (834 bytes)t)
Wave speed on a stretched string with tension tau2.gif (826 bytes) and density mu2.gif (832 bytes) 17-24.gif (186 bytes)
Average power transmitted by a stretched strong 17-35.gif (213 bytes)
Interference of waves

y' (x,t) = [2 ym cos onehalf.gif (67 bytes) phi2.gif (845 bytes)] sin (kx - omega2.gif (834 bytes)t + onehalf.gif (67 bytes) phi2.gif (845 bytes))

if phi2.gif (845 bytes) = 0 the waves are in phase and their interference is fully constructive. if  phi2.gif (845 bytes) = pi2.gif (831 bytes)  the waves are out of phase and their interference is fully destructive.
Standing waves y' (x,t) = [2 ym sin kt ] cos omega2.gif (834 bytes)t
Resonance f = v/lambda2.gif (834 bytes) = n (v/ 2L)     for n = 1, 2, 3, . . .

[Top] [Previously Asked Questions] [References]


Mechanics List of Topics

Measurements Newton's Laws Potential Energy and Conservation of Energy Rotation of
Rigid Bodies
Elasticity
Vectors Forces and Fields Linear Momentum Angular Momentum Mechanical
Oscillations
Motion of Point-Mass Objects in One Dimension The Gravitational Field Collisions Torque Mechanical Waves
Motion of Point-Mass Objects in Two and Three Dimensions Kinetic Energy
and Work
Circular Motion of Point-Mass Objects Equilibrium Sound

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