A study of the sound radiation from musical instruments in rooms using the equivalent source method U. Peter Svenssonl, Mayumi Nakrtno, Kmihiro Sakagami, and Masayuki Morimoto Environmental Acoustics lab, Faculty of Engineering, Kobe University, Nada, Kobe, Japan Abstract This paper studies the sound radiation from a kettledrum, taking the influence of a reflecting surface into account. The time-domain formulation of the method of equivalent sources is used for the sound radiation calCUI ation. Surface layers of monopole sources are used, making the method comparable to a time-iterative boundary element method. For the monopole-like 01 mode, the radiated power can vary ca, +1 dB as caused by the floor reflection, but for the dipole-like 11 mode, the effect was, as expected, much smaller. BACKGROUND The sound field from a membrane with vibration modes, such as on a kettledrum, can be studied analytically for a membrane in an infinite rigid baffle, and a free membrane. Both of these cases can employ the image source method, as in (1). However, such studies do not take the kettledrum body or the room surfaces into account. The effects of these two factors are studied in this paper using a numerical method for the sound radiation. The method of equiva- lent sources with its time-domain formulation is efficient for the calculation of sound radiation (2). Here, a surface layer of monopole sources on the surface of the vibrating body is viewed as a set of equivalent sources, making the method comparable to a time-iterative boundary element method (BEM). The iterative time-domain formulation in (2) leads to very efficient calculations, as has also been observed for the BEM (3,4). Furthermore, to study various vibration patterns, a Green’s function technique is employed, as in (2), The sound radiation is studied separately for each mode, and consequently, the membrane properties and the air cavity’s resonances are not included since they do not affect the membrane mode shapes. CALCULATION METHOD The method of equivalent sources is, like the Kirchhoff-Helmholtz Integral Equation, based on the principle that a set of sound sources reproduce a desired boundary condition (BC) on a closed surface. Here, quadrilateral patches of free-field radiating monopole layers on the surface are used as equivalent sources. Assuming that these quadrilateral patches vibrate with a constant amplitude over the surface, and that the mid-point value of the BC can represent the true BC distribution, the method is essentially the same as the BEM. Using a discrete number of sources, and fulfill- ing the BC at the same number of surface points makes it possible to formulate a discrete-time matrix equation, as- suming that the normal velocity is the relevant BC. This matrix equation is a discrete-time convolution sum, Vn (KT) = ~~$v (kT)q[(K – k) T], [1] where vn(K~ are the values of the (desired) normal veloeity, in the middle of the surface patches, at the discrete time KT, K being an integer, and T= l/f~ where fs is the sampling frequency. q(K~ are the values of the (sought) source signals (= normal velocity) of the patch sources, and hv(K~ are the impulse responses (IRs) from all the source patches to the surface patches. It is assumed that the ~s are no longer than K time samples for any patch-to-patch combination. The discrete-time IR from a source patch to a receiver patch can be approximated as [ ~ (m) dScosp A(fl-R/c) A(KT-R/c)-A((K -l) T-R/c) v 2 ZR +fs R 1 [2] c where A(K~ is a unit pulse, which is a discrete-time representation of the Dirac pulse at). dS is the area of the source patch andR the distance between the midpoints of the source and receiver patches. q is the angle between the direction vector (from source to receiver patch) and the normal vector at the receiver patch. The Dirac derivative ~(t) is represented by two such unit pulses of opposite signs, separated by one time sample. When the source and re- ceiver patches are the same so that R = O, hv(K~ equals A(K~ indicating the identity between a certain patchs source signal, and the normal velocity at the same patch. For patch combinations which are close to each other, the patches must be subdivided to find accurate values of the IRs. This discrete-time representation gives less than 0.5 dB error if the oversampling ratio is at least 20, for Rfl, and no error for R=O. The IRs will typically be 2-6 sam- ples long, plus initiaf zeros, and once these IRs have been calculated for a certain model, any BC can be studied us- ing the same set of IRs. A very important fact is that hv(K~ for many patch combinations will contain initial ze- 1 On leave from Chalmers Univ. of Tech., Gothenburg, Sweden and sponsored by JSPS, Japan. 365