Classical Wave Equation
Before getting to the topic of quantum mechanics I want to introduce you to the classical wave equation. This equation is the one that applies to classical (non-quantum mechanical) waves in nature such as sound waves, light waves, waves on strings, etc. The topic is important since you will certainly see such equations in the future, and it is also a nice introduction to partial differential equations which we find in quantum mechanics.
Let us start with the exercise of finding the speed of a wave on a string under tension. We need to specify the tension , the linear density , and will assume during our derivation that the waves are small perturbations around equilibrium so that we are able to use small angle approximations wherever appropriate.
Linear density is the mass per unit length - in this case, of a string. It is measured in kg/m. The mass of any segment of length L is simply . In the event that the linear density changes in a way that's describable by a function , we need to integrate to find mass, or . While this is true, we will assume linear density is constant during our derivation process. The tension in the string is going to be denoted and is measured in newtons.
Suppose a small disturbance is created on a string under tension. That disturbance can be over a small angle assumed to be circular in shape. If this seems odd to you, consider that a sine wave fits quite well to a circle of carefully chosen radius (an osculating circle) as shown in the diagram below.
Sine Wave with Osculating Circle
Derivation of Wave Speed
Before we derive the wave equation, we will derive the speed of a wave on a string under tension. Based on the idea above, we will use the properties of circular motion to derive the wave speed.
Rather than waves passing by on the string, we will view the system from the reference frame of the moving wave. In that frame the wave stands still and the physical string is moving past. As parts of the string pass over the top of the arc in the diagram, they are undergoing circular motion. As you recall from our mechanics studies, circular motion implies a centripetal force that leads to a centripetal acceleration of magnitude The forces that together make up the centripetal force are the two downward components of the illustrated forces in the diagram. The magnitude of the forces are just the tension in the string. Their downward components are each and there are two of them working together. Thus we can write:
Derivation of the Classical Wave Equation
We use a rather similar procedure with a slightly different diagram to show that traveling waves of the variety that we've used all semester will naturally arise on a string under tension. Consider the diagram below.
Wave Equation Force Diagram
Now our reference frame will be the stationary string rather than the moving wave. The wave will be assumed to be a small perturbation making small angle approximations valid wherever necessary and appropriate. For instance, the length of string between points A and B in the diagram above will be rather than worrying about the fact that it's not horizontal along the x-axis and therefore wondering about doing the trig. If this bothers you, consider the waves we see on strings in practice as when we play a guitar, or when you created standing waves in lab. Even with the biggest wave forms the string is essentially linear to a very good approximation.
The first part of the calculation is just to use Newton's second law as before. This leads to:
This result is called the classical wave equation, and governs the motion of waves of light (EM waves), sound waves, waves on strings, etc. It is easy to see that a wave of the form will satisfy the wave equation by simply plugging in the derivatives. The fact that the functional form of the solution disappears means that it's a solution. Try to do the math to see that it works. Hopefully it doesn't surprise you that waves are described by this function since we spent so much time studying traveling waves.