Solving a Sound Wave Equation in Physics 1: Halliday, Resnick, and Krane

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The discussion focuses on understanding a sound wave equation from Halliday, Resnick, and Krane's Physics textbook. The user seeks clarification on the derivation of an equation related to density and mass, specifically how to transition from one equation to another. They initially struggle with the differentiation process but eventually grasp that the density function, ρ(V) = m/V, leads to the derivative dρ/dV = -m/V². The conversation emphasizes the importance of correctly applying differentiation rules and understanding the relationship between variables. Ultimately, the user successfully resolves their confusion regarding the mathematical derivation.
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Homework Statement:: This is from 5 ed, Physics 1Halliday, Resnick, and Krane. page 428 about sound waves

I have highlighted the equation that I don't understand. How did the author get it? I understand how they get from the middle side to the RHS of the equation, but I don't understand how they get from the first equation to the middle equation.

I don't have any problem with anything before.
Relevant Equations:: Density: Rho = Mass / Volume

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Differentiation.
 
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Orodruin said:
Differentiation.
Is it like this?

dm/dv^2 = - m /(V^2)
so
dp = dm/dv = dv * (-m/(V^2))
 
No. Use ##\Delta y/\Delta x \simeq dy/dx##
 
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Orodruin said:
No. Use ##\Delta y/\Delta x \simeq dy/dx##

Like this?
##d\rho = \frac{\Delta \rho}{\Delta V}dV##
but ##\frac{\Delta \rho}{\Delta V} = -\frac{m}{V^2}##
 
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Afo said:
Is it like this?

dm/dv^2 = - m /(V^2)
so
dp = dm/dv = dv * (-m/(V^2))
Okay, I know why this is false.
It doesn't work since ##\frac{a}{b} = \frac{c}{d}## doesn't imply ## \frac{a}{b^2} = \frac{c}{d^2}.## If that was true, that would mean ##|dV| = |V|## which is obviously false.
 
Do you know how a derivative works?
 
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Orodruin said:
Do you know how a derivative works?
I am still in high school so this is what I know:

1. We can view ##\frac{d}{dx}## as an operator so ##\frac{d}{dx}f(x) = \frac{df}{dx}## is the derivative of ##f## w.r.t. ##x##.

Another way is to view it in terms of differentials:
2. ##dx## and ##dy## are called differentials. The differential ##dx## is an independent variable. The differential ##dy## is defined to be ##dy = f'(x) dx##

Both ways are rigorous.
 
So, let ##x = V##, ##y = \rho(V)## and use 2 to relate ##d\rho## to ##dV##.
 
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Orodruin said:
So, let ##x = V##, ##y = \rho(V)## and use 2 to relate ##d\rho## to ##dV##.
:doh:Ok, I got it!

##\rho(V) = \frac{m}{V}## is a function of volume and ##m## is a constant.

Thus ##\frac{d}{dV} \rho(V) = \frac{d}{dV} \frac{m}{V} = m\frac{d}{dV} \frac{1}{V} = -\frac{m}{V^2}##
 
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