How do we justify the generality of the wave equation?

In summary, the wave equation was derived from applying f=ma to an oscillating string and assuming that the string is nearly horizontal, resulting in the equation ∂2y/∂x2 = 1/v2 ⋅ ∂2y/∂t2. This equation is claimed to be a general result that describes all wave phenomena, but it only applies to waves of small amplitude relative to wavelength. To generalize it for larger amplitudes, the assumption of a nearly horizontal string would have to be relaxed. However, the wave equation is still useful and has solutions that accurately model various physical phenomena. This is because it is based on a linearity assumption, which is fulfilled by many media. However, physical strings
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Sturk200
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We derived the wave equation by applying f=ma to an element of an oscillating string, yielding

∂2y/∂x2 = 1/v2 ⋅ ∂2y/∂t2.

In order to get this result it was necessary to assume that the string in question was nearly horizontal, so that the angle formed by the tension vector and the horizontal axis satisfied the small angle approximation sinα=α. It follows that the equation applies only to waves of very small amplitude relative to wavelength.

However, it is claimed, by my textbook for instance, that this equation is a general result that describes all wave phenomena. How can the derivation be generalized to waves with larger amplitudes if it requires the assumption that the string is nearly horizontal?
 
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At the end of the day, it just happen that the wave equation has solutions that model extremely well a host of physical phenomena (those phenomenon that we call "wave-like" :-) ) no matter how or what "inspired" us to find it (to find the wave equations).

Just like Poisson Equation is extremely useful to Gravitation and Electrostatics.

Sometimes we discover these equations by means of studying basic examples, sometimes deriving them from other assumptions,...

But at the end of the day, what is important is that they work, they are useful (they represent correctly certain features of certain physical phenomena).
 
  • #3
The sinα=α approximation is essentially a linearity assumption. Which really means, the formula you arrive at will hold for all media that are linear. Which is a constraint that a lot of media fulfill (e.g. EM field). It's really just that physical strings stop being linear at some point. So, the wave equation is very general indeed, but you have to know at which point it is no longer applicable.
 
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FAQ: How do we justify the generality of the wave equation?

What is the wave equation and why is it important?

The wave equation is a mathematical formula that describes the behavior of waves, such as light and sound, in a given medium. It is important because it allows us to make predictions about the behavior of waves, which is crucial in many areas of science and engineering.

How is the generality of the wave equation justified?

The generality of the wave equation is justified through experimental evidence and empirical observations. It has been tested and verified in various scenarios, and its predictions have been found to be consistent with real-world phenomena.

What are the assumptions made in deriving the wave equation?

The wave equation is derived from several fundamental principles, including conservation of energy, Newton's laws of motion, and the properties of the medium in which the wave travels. These assumptions are based on our understanding of physics and are continuously tested and refined as new evidence is discovered.

Can the wave equation be applied to all types of waves?

While the wave equation is a general formula, it may not be applicable to all types of waves. For example, it does not accurately describe the behavior of quantum waves or waves in highly nonlinear systems. However, it is still a valuable tool for understanding and predicting the behavior of many types of waves.

How is the wave equation used in practical applications?

The wave equation is used in a wide range of practical applications, such as in the design of musical instruments, earthquake prediction, and medical imaging. It allows us to make precise calculations and predictions about the behavior of waves, which is essential in many technological advancements.

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