What is the concept behind sound waves in my class notes?

[fruitbubbles]

I am going through my professor's notes and I am having a difficult time with his notes on sound waves.

class notes:

First off, I felt like I understood what simple harmonic motion is, but I don't know what the "simple harmonic oscillator moving through air" is referring to. Second, I do not understand where the equation s = s_max cos(kx-wt) comes from, or what in the world the s is supposed to be referring to.

The next part:

I'm again struggling to understand what the s is, so basically anytime an s is mentioned here I have no clue what is going on. Can somebody go through this and explain to me conceptually what is happening?

 

[andrevdh]

As the wave (periodic disturbance) motions through the medium it causes the particles to oscillate
to and fro. The particles are following simple harmonic motion. s is their displacement from the equilibrium
position - the undisturbed position of the particles.

 

[fruitbubbles]

As the wave (periodic disturbance) motions through the medium it causes the particles to oscillate
to and fro. The particles are following simple harmonic motion. s is their displacement from the equilibrium
position - the undisturbed position of the particles.

So is the wave caused by the piston? The piston creates the wave when it applies pressure, and then when it goes back and forth it creates the "oscillation" effect?

Also, in the second picture, when it shows delta s = s1 - s2, does delta s represent the portion of the particles that remained in the equilibrium position?

 

[Nathanael]

You have an element of gas of length $\Delta x$.
When this element is displaced, the right and left ends of this element are not displaced by the same amount.
$\Delta s$ is the difference between how much the left and right ends were displaced.

Hopefully this explains why $\Delta V = A \Delta s$
($A$ is the cross sectional area and $V$ is the volume)

 

[fruitbubbles]

You have an element of gas of length $\Delta x$.
When this element is displaced, the right and left ends of this element are not displaced by the same amount.
$\Delta s$ is the difference between how much the left and right ends were displaced.

Hopefully this explains why $\Delta V = A \Delta s$
($A$ is the cross sectional area and $V$ is the volume)

okay, I have a few questions about that
1. so I understand that mathematically, $\Delta s$ is the difference between how much the left and right ends were displaced..but what is the significance of it? Like what does it represent conceptually? Is that the..total amount displaced?

2. In regards to $\Delta x$, why is there a delta? I usually understand delta to refer to the "change" in something, so if that's just the original length of the gas, why isn't it denoted as simply x instead of as delta x?

 

[Nathanael]

1. so I understand that mathematically, $\Delta s$ is the difference between how much the left and right ends were displaced..but what is the significance of it? Like what does it represent conceptually? Is that the..total amount displaced?

It represents the change in the length of the element of gas. (Which is why $A\Delta s = \Delta V$)

2. In regards to $\Delta x$, why is there a delta? I usually understand delta to refer to the "change" in something, so if that's just the original length of the gas, why isn't it denoted as simply x instead of as delta x?

I believe it is called $\Delta x$ instead of just "$x$" because "$x$" is used to refer to a specific location along the wave.
Look at the equation $S=S_{max}cos(kx-\omega t)$; the x is used to refer to a specific spatial location of the wave.
[[[[[Edit: just like the "$t$" represents a specific "location" in time. And $\Delta t$ would represent a length of time; just like $\Delta x$ represents a length of space]]]]]
That's why they used $\Delta x$ to refer to the length.

If you think about it; it seems to makes sense that "length" can be considered "change in location."

 

[fruitbubbles]

It represents the change in the length of the element of gas. (Which is why $A\Delta s = \Delta V$)I believe it is called $\Delta x$ instead of just "$x$" because "$x$" is used to refer to a specific location along the wave.
Look at the equation $S=S_{max}cos(kx-\omega t)$; the x is used to refer to a specific spatial location of the wave. That's why they used $\Delta x$ to refer to the length.

If you think about it; it seems to makes sense that "length" can be considered "change in location."

Ohh right, because the gas was compressed, so the length changed. For some reason I thought of it as simply having been displaced, but it was displaced AND compressed, which I guess is why the displacement on the right side is smaller...because the gas got compressed? Or something..

 

[Nathanael]

Ohh right, because the gas was compressed, so the length changed. For some reason I thought of it as simply having been displaced, but it was displaced AND compressed, which I guess is why the displacement on the right side is smaller...because the gas got compressed? Or something..

Exactly.

P.S. I edited my last post (in the middle of it) to hopefully make it a little more intuitive as to why they used "$\Delta x$" instead of "$x$"

 

[andrevdh]

A wave is created by displacement or disturbance at one in in a medium.
This displacement or disturbance then propagates further into the medium
due to its elacity. Think of what happens if someone throws a stone into
a pond.

In this case we have a repetative disturbance of the gas - a piston oscillating
to and fro. This disturbance of the gas cause a longitudinal wave motion to
propagate into it that is the gas particles will oscillate to and fro along the direction
of the propagation of the wave.

The theory here considers what happens to a certain portion of gas with cross
sectionnal area A and length Δx and thus volume AΔx. We are especially interested
in how its volume changes ΔV given by AΔs where Δs is the amount by which the
length of the volume of gas changes. s1 is the displacement of the "rear" and s2 the
displacement of the "front" of the volume of the gas (this is the displacement out
of the undisturbed position). If s1 is the same as s2 then the change in the volume
is be zero. If s2 is larger than s1 then the volume have expanded...