**Description**

A tapered loudspeaker arrary has length 1m and emits a tone at 1500Hz. The tone

is being observed 5m away at an angle on the azimuth plane of 30 degrees.

The following MatLab code calculates and plots the following three things:

- the radiation pattern of the loudspeaker array
- the particle displacement in the immediate vicinity of the source
- the particle displacement in the immediate vicinity of the observer

**M-File ** P1a.m

%Constants

%Frequency emitted, speed of sound, angles, wavelength, wave number

f=1500; c=344; b=-.5:(1/200):.5; bpi=b*pi;l=1; wl=c./f; k=2*pi*f/c;

%1a

for i=1:length(bpi)

P0(i)=sinc((l/wl)*sin(bpi(i)));

P1(i)=0.5*sinc((l/wl)*sin(bpi(i)+1));

P2(i)=0.5*sinc((l/wl)*sin(bpi(i)));

P(i)=P0(i)+P1(i)+P2(i);

end

figure(1);

polar(bpi, abs(P(1,:))); axis tight;

title('Radiation Pattern of Tapered Loudspeaker Array');

%1b

[x,y]=meshgrid(0:.01:1, -1.001:.01:1);

sine=abs(y)./sqrt(x.^2+y.^2);

u=1*sine/wl;

r=(x.^2 + y.^2);

dis=(sinc(u)+0.5*sinc(u+1)+0.5*sinc(u-1)).*cos(k*r)./r;

figure(2);

surf(x, y, dis+1);

axis([0 1 -1 1 -10 20])

caxis([-5 5])

title('Particle Displacement in Vicinity of Source');

view(45, 45)

%1c

[x,y]=meshgrid(3.8:.01:4.8, 2.001:0.01:3);

sine=abs(y)./sqrt(x.^2+y.^2);

u=1*sine/wl;

r=(x.^2 + y.^2);

dis=(sinc(u)+0.5*sinc(u+1)+0.5*sinc(u-1)).*cos(k*r)./r;

figure(3);

surf(x, y, 30*dis+1);

axis([3.8 4.8 2 3 -5 5])

caxis([-5 5])

title('Particle Displacement in Vicinity of Observer');

**Plots and Results**

Radiation Pattern

Particle Displacement (Near Source)

Particle Displacement(Near Observer)

**Conclusions**

The loudspeaker array is emitting a tone at a high frequency relative to its length (length >> wavelength = speed of sound/frequency; 1m >> .23m = 344/1500 m). This implies a narrow main lobe and numerous sidelobes. As you can see, the sidelobes are somewhat suppressed due to the array being tapered. As expected, the narrow lobe points straight ahead (0 degrees; azimuth) because there is no delay being swept through the array. The surf plots show the particle displacement in relation to both the souce and the observer. The source plot shows a clear main lobe at location of the array with diminishing side lobes as you go outward towards 0 degrees. There is very little particle displacement outside the main lobe, meaning that there shouldn't be much particle displacement near the observer. Due to the fact that the observer is standing 5m away from the array at an angle of 30 degrees to the side, there is very little particle displacement seen in the vicinity of the observer.

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