Wave equation with no damping

The wave equation expressed in polar coordinates with no damping is given by

For a drum, boundary conditions are:

,

,

,

To prevent a discontinuity, we assume that:

and

Since the displacement of the drumhead must be finite at all points, we further assume that:

,

The analytic solution is:

This is quite complicated. In order to better visualize this result, we isolate a single term,

where is the nth zero of the mth order Bessel function of the first kind.

The function S(r) divides the drum into sections, and F(t) gives the time dependent oscillation of each section. For example, v₂₂(r,θ,t) gives:

The signs in each section correspond to the relative sign of v₂₂(r,θ,t) in that section. The lines are stationary with time.

Below are some animations of different modes.

λ₀₂

λ ₂₁

λ ₂₂

This behavior can be seen experimentally. The boundaries between sections are stationary, so particles placed on the drumhead will collect these boundaries as the drumhead vibrates. To illustrate this, the drumhead was lightly seasoned with pepper, and tapped first in the center and then along the edge.

It is important to notice that these are only individual modes, the motion of a real drumhead will be a linear combination of all modes. Below is an animation of four modes superimposed.

The presence a particular mode is explained by orthogonal eigenfunctions. The initial conditions can be expressed as an orthogonal sum of all possible vibration modes (eigenfunctions). The more similar an initial condition to a particular mode, the more prominent that mode will be.

Striking the drum in the center will only give rise to λ_0n modes, since there is no θ dependence. Striking the drum off center deforms the drumhead in such a way that λ_mn modes are excited.

If the drum is struck very near the edge, there is more energy in high order modes then if the drum is struck near the center. This explains why a drum sounds higher pitched if struck near the edge.

Drum struck in the center

Drum struck at the edge

Wave equation with damping

The wave equation with linear damping can be written

where k is constant. For simplicity, the initial conditions are assumed independent of theta, and the equation becomes

The boundary conditions are the same as above. For the initial conditions, it will be assumed that the drum stick contacts the drumhead, the drumhead depresses, and then springs upward, and the stick leaves the head. The instant the stick leaves the head, t equals zero.

Mathematically, initial conditions are

Using the separation v(r, t) = f(r)T(t), we obtain

where λ² is constant. This results in two related ODEs.

The first ODE is Bessel’s equation, with a solution of

Applying boundary conditions gives

, α_n is the nth zero of the Bessel function, and a is the radius of the drum (15 cm).

The time portion of the solution is:

Notice that damping not only causes the solution to decay with time, it also affects frequency content. The fundamental frequency and envelope function can be used to find the values of k and c.

The final solution is

, , . (AA10)

Where and .

It is interesting to note that a_n is found using the orthogonality of the Bessel function, which is analogous to finding a Fourier series.

The frequency spectrum is . The first few frequencies are listed below.

n	fn (Hz)
1	220
2	507
3	795
4	1083

Below is a plot of the Fourier spectra for the measured and calculated data.

There various possible causes for the presence of extra peaks in the measured Fourier spectrum. Even if the drum is stuck in the center, modes other than λ_0n could be present because of non-uniform tension in the drumhead. However, the additional measured frequencies do not correspond to any eigenvalues. A likely cause is non-uniform tension on the drum head, leading to different regions vibrating at slightly different rates.

The sound generated by the analytic solution can be played.

Vibration of a Guitar String

The vibration of a guitar string is solved by the 1-dimensional wave equation.

As a is defined as the length of the guitar string (in meters), the boundary conditions will be defined as:

The boundary conditions are both equal to zero because the string is fixed at both ends.

Separation of variables gives a solution of

where a = .8 m. The value the constant c can be found by noticing that the fundamental frequency is . The experimental data gives f₁ = 73 Hz, resulting in c = 116.8 . The string will be plucked a distance b from the end. The initial conditions are therefore

For simplicity, we will let d₀ = 1, since this is only scales the magnitude and has no effect on frequency content. With a = .8 m and b = .15, the constants A_n and B_n are:

Observations

· The dampening of an oscillating system not only causes decay with time, the frequency content is also affected.

· Guitar strings have eigenfunctions consisting of sines and cosines.

· Drums have eigenfunctions consisting of sines and cosines multiplied with Bessel functions of the first kind.

· The eigenvalues (and therefore frequencies) of a guitar string are all integer multiples of each other, which gives the guitar string a musical quality.

· The eigenvalues of a drum are not integer multiples of each other, so a drum is not a harmonic instrument. In order for a drum to have a definite pitch (such as a timpani), one vibrational mode must largely dominate the rest.

· For any linear system, the system response will only contain eigenfunctions that are present in the initial condition or forcing function.