What is the mathematical representation of pressure and amplitude in a 1D wave?

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In summary, the amplitude of a 1D wave from a point source can be represented by either a sine or cosine function, and the pressure at a given radius and time is proportional to the amplitude. The wavelength and frequency of the wave remain constant as it travels.
  • #1
chrom68
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Given a wave that begins at a point source that travels outward spherically, i know that for a 1D wave (from the solution of the wave equation) the amplitude at a given radius and time from the point source is:

1) A(r,t) = (1/r)*A(0)*sin(kr-wt) (or is this cosine because we take the real part of the full solution which is of the form e^i*theta = cos(theta) + i sin(theta)??). Many sites on the net use the sine form.

2) I wanted to know what the pressure at a particular radius and time was.
I know it resembles the equation for A(r,t) because they are proportional. Again is it a sine or cosine form of solution?

3) Even though this is a traveling wave, does the wavelength (between peak amplitudes/pressures for example) remain the same as it travels??
 
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  • #2
1) Whether it is sine or cosine is unimportant. Either is fine. It just depends on how the wave started long ago.
2) The wave is a pressure wave, so A should already represent the pressure.
It is the pressure in a sound wave that affects your ear.
The amplitude of movement is 90 degrees out of phase with the pressure, so if one is sine, the other would be cosine.
3) The wavelength and frequency remain the same as n your equation.
 
  • #3


I can confirm that the amplitude of a wave, whether it is a 1D or 3D wave, follows the same mathematical equation. In this case, for a 1D wave, the amplitude at a given radius and time can be represented by A(r,t) = (1/r)*A(0)*sin(kr-wt). This equation is in the sine form because we are taking the real part of the full solution, which is of the form e^i*theta = cos(theta) + i sin(theta).

Similarly, the pressure at a particular radius and time can also be represented by the same equation, as pressure and amplitude are proportional. Therefore, it is also in the sine form.

In regards to the wavelength, it remains the same as the wave travels. This is because the wavelength is determined by the frequency and speed of the wave, which are both constant for a given medium. As the wave travels, the distance between peak amplitudes or pressures may change, but the wavelength remains the same.

I hope this helps to clarify any confusion about the mathematical representation of pressure and amplitude in a 1D wave.
 

1. What is the difference between pressure and amplitude of waves?

Pressure refers to the force exerted by a wave on a surface, while amplitude is the maximum displacement of a wave from its resting position. In other words, pressure measures the strength of the wave, while amplitude measures the size of the wave.

2. How do pressure and amplitude affect the intensity of a wave?

The intensity of a wave is directly proportional to both pressure and amplitude. This means that as pressure and amplitude increase, the intensity of the wave also increases, and vice versa.

3. How can we measure the pressure and amplitude of waves?

Pressure can be measured using a device called a manometer, which measures the force exerted by a wave on a surface. Amplitude can be measured using a device called a seismometer, which measures the displacement of the ground caused by a wave.

4. How does the pressure and amplitude of waves affect their speed?

Pressure and amplitude do not directly affect the speed of a wave. The speed of a wave is determined by the properties of the medium it is traveling through, such as density and elasticity. However, a higher pressure or amplitude may result in a stronger wave that can travel faster.

5. What is the relationship between pressure and amplitude in a wave?

Pressure and amplitude are directly related, meaning that as one increases, the other also increases. This is because the amount of force exerted by a wave is directly proportional to its size or amplitude.

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