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 t  (derived below, the difference in time (t) will give the distance (d) to the epicenter.

This equation (d = 1.13 km/s 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.