Amplitude, Sine or Cosine Wave?

Question

Okay, so in my science class, we're studying waves.

Ex. Mechanical waves, electromagnetic waves, transverse waves, etc...

My group conducted experiments with a slinky (did I spell that correctly?) and we're suppose to demonstrate amplitude, compression, and constructive interference.

What type of wave is it when the slinky slides up and down along the floor? Sine wave, cosine wave, or does it depend on the way we move it?

I know that cosine waves start off with the crest at the y-axis, but I'm not quite sure of it in the real world.... Can't really explain how I don't understand xD!

And this is just something that came up on my interest, but how do I find the amplitude of the following?

y = sin x + 2sin 2x
y = sin 3x + cos 4x
y = cos 6x + cos x

Do I have to use identities?

Thanks.

Answer 1

Common types of waves are longitudinal (compression) and transverse. In longitudinal wave, perturbation is parallel to the direction of propagation (elastic spring). In transverse waves, perturbation is perpendicular to the direction of propagation (guitar string).
http://hyperphysics.phy-astr.gsu.edu/hbase/sound/tralon.html

If you compress and decompress a slinky along its axis, you will get a longitudinal wave. If you shake the slinky, like a rope, you will get a transverse wave.


You can add only waves with the same period. Then you should use usual rules for sines and cosines, e.g.

sin(x) + cos(x) = √2 sin(x+π/4).

The amplitude is √ 2 in this example.

If you add waves with different periods, you cannot get a single harmonic wave. The amplitude of the sum of such waves, e.g. sin(x) + 2 sin(3x), is not defined.

There are some situations, where you can add waves with small diifference in periods. Whether the difference is small or not, is a physical, not a mathematical question. The answer depends on the concrete problem.

Answer 2

There is essentially no difference between a "sine wave" and a "cosine wave." The cosine function is identical to the sine function if you slide it to the right by 90°. Since real waves are moving through space as a function of time, they look like a "sine" function if you look at them at one instant, and a "cosine" function if you look at them slightly later. For that reason, it's an artificial distinction, and generally speaking they're always referred to as "sine waves" (I've never heard the expression "cosine wave").

> how do I find the amplitude of the following?

When the components have different periods (as in your examples), the amplitude of the sum is the sum of the amplitudes.

> y = sin x + 2sin 2x
amplitude = 3 (1+2)

> y = sin 3x + cos 4x
amplitude = 2 (1+1)

> y = cos 6x + cos x
amplitude = 2 (1+1)

Answer 3

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)