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Mathematical description of waves

How do we describe wave propagation mathematically? Typically the wave consists of some disturbance in the medium which depends on space and time. We describe this disturbance as a function
equation50
If this function is to describe a traveling wave then the x dependence at time tex2html_wrap_inline159 must be a shifted version of the x dependence at time tex2html_wrap_inline163. This means that the x,t dependence must take the form :
 equation52
f takes on a specific value for positions and times satisfying
equation55
Therefore the displacement per time unit, in other words the velocity of the wave front is
equation57
An interesting result of Eq. 8 this is that if I look at the spatial dependence of the disturbance at fixed time and the temporal dependence of the disturbance at fixed position I see qualitatively the same functional form though switched around so what comes first with increasing x comes last with increasing t and vise versa. We'll try to apply this to the wave in an audience of people. Imagine we take a photograph of the wave and then examine individuals as a function of increasing x. I'll first see people sitting then people about to sit down then people standing tall then people about to stand up and then people that have not stood up yet. You'll notice that it is like watching a movie of a single individual as the wave passes by him backwards. These considerations also allow us to predict the length of the region where people are standing. In time it takes tex2html_wrap_inline145 to rise then sit which means that in space it takes a distance
equation62

Waves can of course also travel in the opposite direction and then the functional form for the disturbance is :
equation64
since this form describes a shift of a disturbance in the negative x direction with increasing time.


next up previous
Next: The wave-equation Up: Waves Previous: Wave on a taut

Collin Broholm
Mon Nov 10 10:35:21 EST 1997