How Do Pressure and Standing Waves Change with Distance and Interaction?

[physics_06er]

Hi there

Could someone please help with these questions.

A helmholtz resonator has a resonant freq. of 100Hz. draw graphs showing how the pressure and displacement of the air in the neck vary with time. how would these be different at the position of a listener some distance away...

i can draw the graphs but I'm unsure of the part about the listener being some distance away...??

and the 2nd one is ...explain how a standing wave can be formed by traveling waves, and hence how the frequencies of standing waves on a stretched string are related to the speed at which waves travel along the string?

all i know is that waves would be traveling in opposite directions..canceliing out..? is this even right?...any help much appreciaited

Thanks
physics_06er

 

[berkeman]

-1- The resonator will have a partial standing wave in it (partial because some of the sound energy has to get out the open end to propagate to the listener.

-2- What is the equation for a traveling wave? What do you get if you combine a left-travelling wave and a right-travelling wave? Also, what is the relationship between the velocity of propagation of a wave on a string, in comparison to the tension of the string and the linear mass density of the string?

 

[kyle8921]

Sure, I'd be happy to help with these questions about pressure and standing waves.

First, let's start with the helmholtz resonator. This is a type of acoustic resonator that consists of a hollow chamber with an opening, known as the neck, and a volume of air inside. The resonator has a specific resonant frequency, which is the frequency at which it will vibrate most strongly in response to sound waves.

When a sound wave with a frequency of 100Hz enters the resonator, it will cause the air inside to vibrate at that frequency. This vibration will cause fluctuations in pressure and displacement of the air molecules in the neck of the resonator. The pressure and displacement will vary sinusoidally with time, meaning they will increase and decrease in a regular pattern. The specific shape of the graphs will depend on the amplitude and wavelength of the sound wave, as well as the geometry of the resonator.

Now, let's consider the listener who is some distance away from the helmholtz resonator. As the sound wave travels away from the resonator, it will spread out and decrease in amplitude. This means that the pressure and displacement of the air molecules at the listener's position will be lower than at the neck of the resonator. Additionally, the sound wave may also reflect off of surfaces and interfere with other sound waves, causing further changes in the pressure and displacement at the listener's position.

Moving on to the concept of standing waves, these are formed when two identical waves with equal amplitudes and frequencies travel in opposite directions and interfere with each other. This interference can result in areas of constructive interference, where the amplitude of the wave is increased, and areas of destructive interference, where the amplitude is decreased. The points of constructive interference are known as nodes, while the points of destructive interference are known as antinodes.

On a stretched string, the standing wave will have nodes at the ends of the string and an antinode at its center. The frequency of the standing wave is related to the speed at which the waves travel along the string by the equation f = v/2L, where f is the frequency, v is the speed, and L is the length of the string. This means that as the speed of the waves increases, the frequency of the standing wave also increases.

I hope this helps to clarify these concepts for you. Let me know if you have any further questions.