Sine/Cosine Wave?

Question

The instantaneous rate of change for a sive wave like:
a sin (bx + k) is ab cos (bx + k)
The instantaneous rate of change for a cosine wave like:
a cos (bx + k) is -ab sin (bx + k).

So here's my question. that is only the rate of change for the function. BUT!!! For an energy wave, what is the rate of change for the speed of an energy wave? Is it the absolute value of the derivative?

I believe I should rephrase my question.

An energy wave is simplify just a wave with energy.

The energy wave travels in a trajectory like the sine wave and cosine wave, but the derivative only gives the rate of change for the function. Speed cannot be negative, so what is the rate at which energy is moving in the wave?

Answer 1

D.L., it's difficult to understand what you want to know. I guess, your question is about the speed, with which waves propage?

You need to include time and consider the equation of the form f= A exp( -i ω t + i k x). Here, A is the wave amplitude, ω is frequency, and k is the wave vector. Note, that the wave energy is proportional to squared amplitude |A|^2. If you do not like complex numbers, write down f = A sin(-ω t + k x). Then you can determine the wave velocity C:

C = ω/k.

It can be positive or negative depending on the direction of propagation. Also, you should be careful with this equation.
It gives you the so-called phase velocity. The energy (physical signals) is transferred with the group velocity C_{group}. It is equal to the derivative of the function ω(k),

C_{group} = d ω / dk.

Differentiating the value sin (k x) does not give you much information. You see only how the wave phase changes along x, nothing more.

If this does not answer your question, let me know.

Answer 2

No it is directly the derivative of the energy function
Ex if the electric circuit is purely resistive and the voltage is a sine wave the power( not energy which needs integration )
is W = a/R sin^2(bx+k) and the rate of change is
dW/dx = 2ab/R sin(bx+k) cos(bx+k) = ab/R sin 2(bx+k)

Answer 3

What is an energy wave? I know electromagnetic waves, elastic waves, surface waves, gravity waves etc.. I have never heard of an energy wave. Maybe except on Star Trek.

If you want to describe a wave that is propagating in a medium, you need a description that contains a space vector x and a time t. Where is t in your equations? Speed is related to change in time. No change in time, no speed.

A simple plane wave described by e.g.

u(x,t) = a exp(i(kx - wt))

where k is the wave vector (normal on the wave front). Details can be found in

http://en.wikipedia.org/wiki/Plane_wave

Answer 4

by way of fact fact is 3-dimensional. For the math, this is achieveable to map it in an x-y airplane. in spite of the undeniable fact that, in fact, capability consequences although is interior the area next to it in all available dimensions. this is lots like, "why do no longer ants build a nest in one airplane like they do in ant farms?"

Answer 5

im a professor at cambridge university u know the place where stven hawkins teaches at u wrote ur expression wrong