# About acoustics physics -- The Wave Equation and diminishing sound intensity

• Nunzio Luigi
In summary, the conversation discusses the equation $$\frac{\partial^2\psi}{\partial t^2}=c^2 \nabla^2 \psi$$ which describes pressure and propagation speed in acoustics. It also discusses the relationship between sound intensity, power, and distance, using the equation $$I = \frac{W}{4\pi r^2}$$ and the concept of inverse square-law. The conversation is seeking help in understanding how to calculate the necessary power output of a sound diffuser to achieve a desired intensity level.
Nunzio Luigi
Hello everyone! :-)
Actually I'm starting to understand acoustics physics and I figured actually out about this equation:

$$\frac{\partial^2\psi}{\partial t^2}=c^2 \nabla^2 \psi$$

which describes practically about pressure and propagation speed into space and time. I know also this equation describes practically also the decrement of sound intensity in time from a source to a destination...if we would talk about particle pressure it's decrement of pressure in space by inverse square-law.
So knowing, for spherical waves , the sound intensity in a certain point of time is:

$$I = \frac{W}{4\pi r^2}$$

and supposing to have a sound diffusor with max power output of 150W and knowing human ear voice range audibility is about 40dB-60dB and supposing I want to have I = 50dB at the time entering in my ear so how I can calculate which power output I have to set the sound diffusor to obtain that intensity I I said before?

Could you help me with this little example so I can understand and study all steps to obtain all values in all situations?

Thanks in advance to all!
Cheers,
Nunzio Luigi

<Moderator's note: LaTeX fixed. Please see https://www.physicsforums.com/help/latexhelp/>

Last edited by a moderator:
The rms acoustic pressure at 94 dB SPL is 1 Pa. The acoustic intensity (far from the emitter) depends on the acoustic pressure as:##\it I=\frac{p^2}{\rho c}##, where ##\it\rho## is the density and ##\it c## the speed of sound. Now you can find different relationships between acoustic power, intensity, distance and SPL.

## 1. What is the Wave Equation in acoustics physics?

The Wave Equation is a mathematical representation of the behavior of sound waves in a medium. It describes how sound waves propagate through a medium by taking into account factors such as frequency, wavelength, and speed of sound.

## 2. How does the Wave Equation relate to diminishing sound intensity?

The Wave Equation predicts that as sound waves travel through a medium, their intensity decreases due to factors such as absorption, scattering, and dispersion. This is known as diminishing sound intensity and is an important concept in acoustics physics.

## 3. What factors affect the behavior of sound waves according to the Wave Equation?

The Wave Equation takes into account several factors that affect the behavior of sound waves, including the properties of the medium (such as density, elasticity, and temperature), the frequency and wavelength of the sound waves, and any obstacles or boundaries present in the medium.

## 4. How is the Wave Equation used in practical applications?

The Wave Equation is used in a variety of practical applications, including designing acoustic systems, noise control, and analyzing the sound quality of musical instruments. It is also used in medical imaging techniques such as ultrasound.

## 5. Are there any limitations to the Wave Equation in describing sound waves?

While the Wave Equation is a useful tool for understanding the behavior of sound waves, it does have some limitations. It assumes that the medium is homogeneous and isotropic, meaning that it has the same properties in all directions. In reality, most media are not completely uniform, which can affect the accuracy of the predictions made by the Wave Equation.

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