Confused with mechanical waves

Woopydalan

Maybe its just a mesh of putting all these ideas together. Often the problems have some small twists to them, and I get confused putting the pieces together to solve the problem.

Simon Bridge

You are thinking of standing waves in different conditions - yes, they come from the boundary conditions when solving the wave equation. eg.
http://hyperphysics.phy-astr.gsu.edu...es/wavsol.html

When you vibrate a string (say), you get two travelling waves going in opposite directions and reflecting from the boundaries. The waves will destructively interfere and you get nothing or a ind of mess, except at particular speeds, where they reinforce to give the orderly harmonics you have seen.

You get nodes at the points of maximum destructive interference of the two waves - this will be automatic for the fixed ends of a string because the wave inverts on reflection there.

If the end is not fixed, though, the wave does not invert at reflection and you get an antinode there.

It is possible to have a situation where the string in not fixed, but not completely free to move ... and you get less well defined behavior. It is probably a good idea to get used to the simple situations before tackling that.

A theme throughout has been that you have been thrown by the similarity in the math for different situations ... this can happen when you have been used to a problem-solving method that requires that you "find the right equation" and then plug numbers into it.

You have to move away from that - let the physics dictate the form of the equation, then the math becomes a language.
Keep the physics in mind and these confusions go away.... and you don't have to memorize so many almost-alike equations ;)

Woopydalan

I don't know why in simple harmonic motion the displacement x is not a factor, whereas in wave motion it is. Is this because with SHM, the spring is translating only back and forth, so it is moving in one direction, so you can't have two types of locations in space affecting the equation?

With the string vibrating, the wave is propagating forward, yet the elements on the string go up and down, so there are two components of location involved, hence two location parameters involved in the equation? the x referring to where the element is on the string and the y to mean where the wave is at that particular element (does the Y imply its vertical displacement?)

With sound waves, the air particles are moving longitudinally, essentially vibrating back and forth, so I don't see why you can have two parameters of location involved, since there is no vertical displacement.

Simon Bridge

Like for the horizontal mass-on-a-spring you mean ... in which case x(t)=x(0)+Asin(ωt+δ) ... where x(0) is the x location of the equilibrium position.

Normally you'd put y(t)=x(t)-x(0)

That would be SHM. Is it a solution to the wave equation?

Remember the variable y can be anything - it does not have to be a transverse displacement.

For the sound wave ... x(t) itself depends on x... which can get confusing:
We'd define y(t) the same way and write y(t)=Asin(kx-ωt+δ)
(IRL we describe sound waves in terms of air pressure.)

Now - what was your question?

Woopydalan

Ok so first you said to think of the physics and forget the math, then when I try to understand the physics you use the math as your safety net of explanation :P

For sound waves

S(x,t) = S_max cos(kx - ωt) (Why is the phase angle omitted??)

Imagining a piston oscillating back and forth, it pushes air molecules toward the end of the tube, when it pushes the batch of air, there is less air density at that area when the piston returns back to its initial position, so the air molecules return back to fill that void, hence oscillations of the batches of air. The original air pushes another batch of air, then the air ensity in that spot is less so the 2nd batch returns, and that behavior continues.

This situation is what s is referring to, yes? Such that the S_max is the amplitude of the oscillations of the individual batches of air. The x is refering to where along the line in space that batch of air molecules is present? And S(x,t) is refering to where in that individual range that the air molecule batch can oscillate? So it makes sense that the batches translate and oscillate on the same axis, so S(x,t) is the position along that x axis. I think this makes sense to me, is it right? I can see why the position x is important in determing where a certain batch of air located on the x axis is at a certain time in its oscillation

Woopydalan

Also, why is it that sinusoidal waves are used to represent sound, when it is longituinal and not transverse?

Is the difference between a standing wave and a non-standing wave boundary conditions?

Simon Bridge

I probably also said to use the math as a language ;)
I suspected that you may have been taking the use if y to literally mean a transverse displacement and I wanted to show you that it can mean anything I want.

[edit]It seems what I actually said was:... moving away from a "find the right equation" approach to problem solving is not the same as "forget the math".

It isn't - the relative phase angle is 0 for this particular example if the general equation is y(x,t)=Acos(kx-ωt+δ) and π/2 if the general equation is y(x,t)=Asin(kx-ωt+δ).

The phase angle, or just "phase" for short, is θ=kx-ωt+δ ... the δ is the relative phase for the situation that S(0,0)≠S_max.

There are lots of ways to make an oscillating pressure wave and that would be one of them, yes.For a particular molecule, x is it's equilibrium positionIt tells you how far from it's equilibrium position a molecule normally at position x has been shifted.

Easier to understand if S is pressure ... a pressure gauge at position x will show a reading of S(x,t).

Woopydalan

Was my interpretation of s(x,t) incorrect? I'm thinking the equation must be describing where a particular batch of air is at a certain time.

So x is the equilibrium position of the batch of air, so once it is disturbed and is displaced, it is no longer in equilibrium. Knowing the equilibrium position x and its displacement s(x,t) on its oscillation, we can determine where that particular batch of air is at any given time?

Simon Bridge

It was OK ... The x part is where it came from, the s part is where it went and the t part is how it got there.

This is not usually a useful description of sound though ... you cannot tell from looking at it where a particular bit of air came from - you only know where it is and where it is going... which may or may not mean something. But you can watch the air pressure and how it varies about some average.

Woopydalan

Well usually I can translate the motion of the air molecules into pressure differences. If the air molecules oscillate a large distance (i.e. more air is compressed), the difference in pressure should be higher. Right?

Simon Bridge

Difference in pressure - yes ... the maximum pressure difference would be twice the amplitude.

Excuse me it's a habit - in physics we always have to bring everything back to something that can be measured. We can sometimes physically track the displacement of air, sort of, by watching dust suspended in it. When we do that we find the actual motion of individual molecules is quite complicated even when the wave motion is simple.

It's easier to see this with water because you can see the waves on the surface (obvious up-down motion right?) and you can float small bits of neutral buoyancy stuff in it to see how the local bits of water are moving. What you find is that the actual water moves in a circle while the waves go up and down.

I think what you need to do now is look for lots of wave demonstrations online - and set up some waves yourself.