Consider a drum head constructed by stretching a membrane over a stiff frame that encloses a flat 2D domain. The vibration of the membrane is described by the wave equation Helmholtz equation with the Dirichlet boundary condition at the periphery of the domain where the membrane is constrained by the stiff frame. In this case, there is a set of discrete solutions to the wave equation, called normal modes or eigenmodes , each of which vibrates at a characteristic frequency, called eigenfrequencies.
The lowest eigenfrequency defines the fundamental tone, which for instance could be concert pitch A Hz. The set of higher eigenfrequencies, or overtones in musical terms, gives rise to the tone color or timbre of the vibrating membrane. Is it possible to construct two drum heads with different shapes that share a set of eigenfrequencies? The idea was that if the two drums have an identical set of eigenfrequencies being isospectral , then they would have the same timbre and sound the same to the ear, even though their shapes are different.
Kac commented on the asymptotic behavior of the eigenfrequencies in the limit of very high frequencies and made connections to various branches of physics and mathematics to provide a ground for intuitive understanding.
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The eigenvalues of the two polygons can be computed numerically, which is shown in this Isospectral Drums model in our Model Gallery. The image below shows the first three normal modes of the two polygons that share the same set of eigenfrequencies:. In other words, they expected that, in general, one can hear the shape of a drum, unless the shape of the drum is specially constructed to be isospectral with another shape, like the two polygons depicted above. In the following discussion, we will take a closer look at such special shapes by considering various physical mechanisms involved in the sound production and detection.
We will find that when we include relevant physical effects, we actually can tell two drums apart by the sound, even if they are specially constructed to share the same set of eigenfrequencies.
The first effect we will examine is the excitation of the vibrational modes in the membrane. Since the timbre is determined by the set of relative amplitudes of the normal modes, it is not enough to just have an identical set of eigenfrequencies for the two drums to sound the same. They also need to have the same relative amplitude for each eigenmode, which may not be trivial to achieve. Each location of striking is somewhere in the middle of the drum, where a child may instinctively choose to hit if given such a drum and a drum stick.
We use COMSOL Multiphysics simulation software to calculate the frequency response of each of the locations and plot the results in the graphs below. We first focus on just one drum, say, the one on the left. As we hinted at earlier, the drum sounds differently depending on the location where it is struck by the drum stick.
We see different energy distribution among the first three eigenmodes, which will result in different timbre. This is, of course, a well-known fact to percussionists and is the result of the same principle that enables a single bell to ring in two distinct tones, as demonstrated by this ancient set of bells from over two thousand years ago. Is there any hope that we can make the two different drums sound the same?
As we examine the graph, it becomes evident that none of the dashed curves match the solid curves in all three of the eigenmodes.
In other words, the two drums do sound differently, even though they are isospectral, sharing the same set of eigenfrequencies. However, this simple example illustrates that it is not an easy job to make the drums sound the same, due to the different coupling strengths of energy from the drum stick to the various vibrational modes of the membrane.
And, none of these drums will produce a sound with a definite pitch. I hope this will arouse the interest of mathematicians who are reading. In other words, they expected that, in general, one can hear the shape of a drum, unless the shape of the drum is specially constructed to be isospectral with another shape, like the two polygons depicted above. An elegant analytical solution similar to those shown in the papers mentioned above would be much nicer. The shaman uses incantation, drums, rattles made of membranes, wood and metal, or gourds wrapped with beads to change human brainwave functions. The Master experiences the drumming.
The magic of mathematics never ceases to amaze us. Not long after the two isospectral polygons were published, Buser, Conway, Doyle, and Semmler constructed a pair of domains that are not only isospectral sharing the same set of eigenfrequencies , but also homophonic: In other words, if the special point of each drum is hit by a drum stick, then each corresponding pair of eigenmodes of the two drums will be excited with the same amplitude and the two drums will sound the same.
In the following graph, we plot the computed frequency response of the two drums to a narrow Gaussian area load centered on each of the special points:. Our ears do not sense the vibration of the membrane directly. Rather, the sensing is mediated by the acoustic wave in the air.
In this case, we can easily compute the frequency spectrum of the sound wave using COMSOL Multiphysics to find out what we really hear with our ears. For convenience, we will call them the first, second, and third mode, with the understanding that there are other modes in between being neglected since they are much less energetic.
The polar graph below compares the computed sound pressure level in dB in the plane of each of the two drums, a few meters away from each drum.
We see that the sound pressure field produced by the first mode is more or less independent of direction solid and dashed blue curves. This is not surprising, since the mode shape of each drum looks pretty much like a monopole source:. On the other hand, the directionality of the sound field from the second or the third mode of each of the drums is quite pronounced and also quite different between the two drums. For example, for the second mode, the sound field from Drum 1 looks like a dipole field solid red curve , while the one from Drum 2 is more complex dashed red curve.
This observation again matches what we see in the mode shapes of the two drums:. What really determines the perceived timbre is the ratio of the amplitudes of the higher modes the overtones to the lowest mode the fundamental tone. So, in the next graph, we plot the amplitude ratios of the second and the third modes to the first mode, at a sampling of directions:.
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The pool is up to the wall at West-gate. The end comes soon. We hear drums, drums in the deep. Sign In Don't have an account? Retrieved from " http: