Deriving equation of wave motion

In summary, the equation (2.9) described in Chapter two of Eugene Hecht's optics book is a linear differential equation that also explains wave phenomena. However, it is not considered a wave equation because it is a simple and general equation that can describe any differentiable function. According to the text on page 21, a second order differential equation is required to fully describe a wave, as it involves two parameters (amplitude and frequency) which require two conditions. The derivation of the D'Alambert equation in this book is criticized for being the worst seen, despite its simplicity.
  • #1
Pushoam
962
51
The equation below (2.9) is also a linear differential equation.
This equation also describes the wave phenomena.
So, why is this equation not considered as wave equation?
I have taken it from the optics book by Chapter two Eugene Hecht,5th edition ,Pearson.
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391fdd9f-40f6-4648-874e-37d04b73169a
 
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  • #2
I must say that this is the worst derivation of the D'Alambert equation I've ever seen. It's simple but seems to describe any differentiable function. It is true that "wave" is a really general concept, in fact there are many different kind of waves and many different equation describing them, but it's not like anything is a wave!
Anyway, as the text say on page 21, you need at least a second order differential equation to describe a wave since it has in the simplest case an amplitude and a frequency, so two parameters which requires two conditions.
 

1. What is the equation of wave motion?

The equation of wave motion is a mathematical representation of the physical properties and behavior of a wave. It describes how a wave travels through a medium, such as air or water, and how its amplitude and frequency change over time.

2. How is the equation of wave motion derived?

The equation of wave motion is derived using the principles of wave mechanics, which involve the study of the motion and behavior of waves. It is based on the concept of wave propagation, where the wave travels through a medium by transferring energy from one point to another.

3. What factors affect the equation of wave motion?

The equation of wave motion is affected by several factors, including the properties of the medium through which the wave is traveling, the frequency and amplitude of the wave, and any external forces acting upon the wave. These factors can alter the shape and speed of the wave as it propagates through the medium.

4. What are the important variables in the equation of wave motion?

The important variables in the equation of wave motion include the speed of the wave, the wavelength, and the amplitude. These variables determine the characteristics of the wave, such as its energy, frequency, and direction of propagation.

5. How is the equation of wave motion used in real-world applications?

The equation of wave motion has many practical applications, such as in the fields of acoustics, optics, and seismology. It is used to study and understand various phenomena, such as sound waves, light waves, and seismic waves, and to design and develop technologies and devices that utilize wave motion, such as sonar, lasers, and earthquake detection systems.

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