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

In summary, the conversation is about understanding an equation from page 428 on sound waves in a Physics textbook. The equation involves density and differentiation, and the author is trying to understand how it was derived. The conversation also includes a discussion on derivatives and differentials, with the final conclusion being that the equation can be understood by using the fact that density is a function of volume and the derivative of this function is equal to -m/V^2.
  • #1
Afo
17
5
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

1639119782840.png
 
Physics news on Phys.org
  • #2
Differentiation.
 
  • Like
Likes Afo and vanhees71
  • #3
Orodruin said:
Differentiation.
Is it like this?

dm/dv^2 = - m /(V^2)
so
dp = dm/dv = dv * (-m/(V^2))
 
  • #4
No. Use ##\Delta y/\Delta x \simeq dy/dx##
 
  • Like
Likes Afo
  • #5
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}##
 
  • Like
Likes vanhees71
  • #6
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.
 
  • #7
Do you know how a derivative works?
 
  • Like
Likes Afo
  • #8
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.
 
  • #9
So, let ##x = V##, ##y = \rho(V)## and use 2 to relate ##d\rho## to ##dV##.
 
  • Like
Likes Afo
  • #10
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}##
 
  • Like
Likes vanhees71, Orodruin and TSny

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

1. What is a sound wave equation?

A sound wave equation is a mathematical equation that describes the behavior of sound waves in a given medium. It relates the physical properties of the medium, such as density and elasticity, to the characteristics of the sound wave, such as frequency and wavelength.

2. How do I solve a sound wave equation?

To solve a sound wave equation, you will need to use mathematical techniques such as algebra, trigonometry, and calculus. You will also need to understand the physical principles behind the equation, such as the relationship between pressure and density in a sound wave.

3. What is the role of the Halliday, Resnick, and Krane textbook in solving sound wave equations?

The Halliday, Resnick, and Krane textbook is a widely used resource for learning about physics, including sound waves. It provides clear explanations and examples of sound wave equations, as well as practice problems and solutions to help you improve your problem-solving skills.

4. Can I apply the sound wave equation to real-world situations?

Yes, the sound wave equation can be applied to real-world situations, such as understanding the behavior of sound waves in different mediums, predicting the frequency and wavelength of a sound wave, and determining the speed of sound in a given medium.

5. Are there any limitations to the sound wave equation?

Like any mathematical model, the sound wave equation has its limitations. It assumes ideal conditions and may not accurately describe the behavior of sound waves in all situations. It also does not take into account factors such as interference and absorption, which can affect the behavior of sound waves in the real world.

Back
Top