musical instruments. • The normal modes a string : lutes. • The normal modes of an uniform membrane : drums. • The normal modes of a non-uniform membrane : Indian drums. • Can one hear the shape of a drum ? The isospectral problem. • The numerical synthesis of sound.
(1878) in Brussells. • Idiophones : the body of the instrument vibrates. • Aerophones : the air enclosed by the instrument vibrates. • Chordophones : the string of the instrument vibrates. • Membranophones : the membrane of the instrument vibrates.
classic simplicity to obtain the normal modes of the mridangam. He found that the ﬁrst ﬁve overtones were integer multiples of the fundamental. We will reproduce Raman’s study using modern methods in the afternoon demos.
= 2.9582 h = 2.9582 h = 3.0535 h = 3.9541 h = 3.9541 h = 4.0204 h = 4.0204 h = 4.9041 h = 4.9687 h = 4.9687 h = 4.9727 h = 4.9727 h = 5.7441 h = 5.7441 h = 5.9409 h = 5.9409 h = 5.9897 Nodal lines
= 1.3843 h = 1.5617 h = 1.5628 h = 1.7341 h = 1.7752 h = 2.0903 h = 2.1012 h = 2.1017 h = 2.2158 h = 2.2166 h = 2.4850 h = 2.4934 h = 2.5593 h = 2.5613 h = 2.7926 h = 2.7928 h = 2.8490 Nodal contours : eccentric
= 2.9582 h = 2.9582 h = 3.0535 h = 3.9541 h = 3.9541 h = 4.0204 h = 4.0204 h = 4.9041 h = 4.9687 h = 4.9687 h = 4.9727 h = 4.9727 h = 5.7441 h = 5.7441 h = 5.9409 h = 5.9409 h = 5.9897 Nodal contours : concentric
1960) If the eigenvalues of a membrane bounded by an arbitrary region is given, can one infer the shape of the region from the eigenvalues ? Answer is no! Gordon, Webb, Wolpert, 1991