where is matrix form of FDTD method, is state vector, is difference operator, is identity matrix and is zero matrix.

Simulation conditionThe size of sound field 10 m × 10 mShape of initial valueRadiusSize of initial value vector

Disk1 m997

Shape of observation pointsRadiusNo. of observation points

Circle1.1 m

60Sound speed 340 m/sDensity 1.21 kg/m3

Spatial discretization interval 0.1 mTime discretization interval 1/48000 s

From propagated waves generated by estimated initial value, itcan be confirmed that directional sound sources ware realized byproposed method. It can be seen that manually setting such aninitial value is not an easy task even when the desired directivity isa simple pattern.

Estimated initial value corresponding to directivity (b)

Formulation with matrix form of FDTD methodThe least squares formulation with matrix form of FDTD methodis written as following:

where is expanding operator and is observation operator.There are following relation:

Propagated waves generated by estimated initial value and their directivity

REALIZING DIRECTIONAL SOUND SOURCE IN FDTD METHOD BY ESTIMATING INITIAL VALUE

Daiki Takeuchi, Kohei Yatabe, Yasuhiro Oikawa (Waseda University, Japan)

Example of sound pressure at the observation point which constructs a

IntroductionFinite-difference time-domain (FDTD) method is one of the mostpopular wave based acoustical simulation methods owing to itsstraightforwardness of calculating an impulse response. In anFDTD simulation, initial value of sound field decides the behaviorof the wave propagation because FDTD methods have recursivescheme. However, most of FDTD simulations have beenperformed with an omnidirectional sound source which is notrealistic because the real sound sources often have directivities.

In this paper, a method of imposing a directional sound source intoFDTD methods is proposed. By estimating initial value from desireddirectivity pattern, the directivity is approximately realized.

In the proposed method, the initialvalue is assumed to be compactlysupported on a small region anddirectivity is evaluated at observationpoints. Therefore, estimating initialvalue problem is formulated as

Matrix form of FDTD methodFor solving this problem easily, matrix form of FDTD method isintroduced. FDTD method can be rewritten as following matrixform:

Formulation estimating initial value problemIn an FDTD simulation, initial value of sound field decides thebehavior of the wave propagation. Therefore, the initial valuewhich generate directional sound wave propagation can beestimated from a wave with the desired directivity pattern which isideal created from the given pattern by multiplication

ConclusionsThe method of realizing directional sound source by estimatinginitial value of an FDTD method was proposed. The proposedmethod can approximately impose any directivity pattern in FDTDmethod. Future works are investigation of the effect of the settingof the region of the initial value and observation points andanalysis of the property of the matrix .

To confirm appropriatenessof the proposed method, anumerical experiment wasconducted. For solving leastsquares problem, LSMRsolver was utilized. In sucha iterative solver, explicitconstruction of matrix is notnecessary because theresult of matrix vectorproduct only required.

Numerical experiment

where is initial value vector, is operator of FDTD methodand is ideal wave propagation at observation points, respectively.