Waves under Boundary Conditions

In summary, waves on a string under boundary conditions that are not at a harmonic frequency are similar in form to stable standing waves, but are not as resonant. The modes of the waveguide that arise due to the boundary conditions represent the possible solutions of waves that can travel through the waveguide, but frequencies between these modes cannot be a solution and are instead evanescent modes. This is due to the fact that the wave on a string only has one dimension of freedom and frequencies between two modes cannot be a solution. If the oscillator were continuously running, it would create forced oscillations all across the wave, but these waves would still not be stable enough to exist for the one dimensional case and would eventually die out due to attenuation.
  • #1
Gear300
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For a string with one endpoint attached to a wall and the other to an oscillator (so that it is under boundary conditions), what is the character of waves that are not at a harmonic frequency?
 
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  • #2
I was assuming that the waves would be somewhat similar in form to the stable standing waves (considering the forced nodal points), but not as resonant (out of phase from the nearest resonance frequency).
 
  • #3
If we had a rectangular waveguide, we could solve for the modes of the waveguide that arise due to the boundary conditions. These modes represent the possible solutions of waves that can travel through the waveguide. But since these modes are discrete, it can be a bit confusing on how waves of frequencies between two modes can propagate. What happens is that we discretize the possible wave numbers in say the x and y directions and allow the wave number in the z direction to be continuous. So as we change the frequency, the direction of the wave as it propagates changes slightly so that it bounces around while satisfying the boundary conditions.

The problem with the wave on a string as I see it is that we only have one dimension of freedom, along the length of the string. Thus, we do not have an extra dimension that we can vary freely to allow continuous solutions to the eigenvalue problem. That is to say, if we have two frequencies of waves that satisfy the boundary conditions (driven at one end and a nodal point at the other), then frequencies between those two modes cannot be a solution. Instead, they are evanescent modes, if you were to excite these modes on your string they would die out as they travelled.

You could show this explicitly by solving for the wave equations and you should arise with one that attenuates over time/distance.
 
  • #4
I see...so altogether, they are not stable enough to exist for the one dimensional case...though, if the oscillator were continuously running, would it sort of be the same as the condition for forced oscillations all across the wave?
 
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  • #5


When a string is under boundary conditions, the waves that are not at a harmonic frequency will exhibit a non-uniform pattern. This is because the string is constrained at one end, causing a reflection of the wave back towards the oscillator. This reflected wave will interfere with the original wave, resulting in a complex waveform with varying amplitudes at different points along the string. This non-uniformity is in contrast to the regular, periodic pattern seen in waves at harmonic frequencies. Additionally, the amplitude of the waves will decrease as they travel towards the fixed end due to energy loss through the reflection process. Overall, the character of waves under boundary conditions can be described as non-uniform and dissipative, as opposed to the regular and sustained nature of waves at harmonic frequencies.
 

1. What are boundary conditions for waves?

Boundary conditions for waves refer to the specific conditions that must be satisfied at the boundary or interface between two mediums in order for a wave to propagate through them. These conditions can include factors such as the medium's density, elasticity, and velocity.

2. How do boundary conditions affect wave behavior?

Boundary conditions play a crucial role in determining how a wave will behave as it moves from one medium to another. These conditions can affect the amplitude, frequency, and wavelength of the wave, as well as its speed and direction of propagation.

3. What is the difference between open and closed boundary conditions?

Open boundary conditions refer to a boundary between two mediums that allows for the transmission of waves without any reflection. This is often the case for waves traveling through air or water. Closed boundary conditions, on the other hand, refer to a boundary that reflects waves back into the same medium, such as a wave reflecting off a solid wall.

4. How are boundary conditions used in practical applications?

Boundary conditions are used in various practical applications, such as in seismology to study earthquake waves, in acoustics to design soundproofing materials, and in optics to understand the behavior of light waves at the interface of different materials. They are also critical in engineering and design, particularly in the development of materials and structures that can withstand the effects of waves.

5. Can boundary conditions be manipulated or controlled?

In some cases, boundary conditions can be manipulated or controlled to alter the behavior of waves. This is known as boundary condition engineering and is commonly used in fields such as optics, where the design of materials and structures can be optimized for specific wave behaviors. However, in many cases, boundary conditions are inherent to the properties of the mediums involved and cannot be easily changed.

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