A soprano and a bass are singing a duet. While the soprano sings an A-sharp at 932 Hz, the bass sings an A-sharp but three octaves lower. In this concert hall, the density of air is 1.20 kg/m and its bulk modulus is 1.42 X 105 Pa. In order for their notes to have the same sound intensity level, what must be (a) the ratio of the pressure amplitude of the bass to that of the soprano and (b) the ratio of the displacement amplitude of the bass to that of the soprano? (c) What displacement amplitude (in m and in nm) does the soprano produce to sing her A-sharp at 72.0 dB?
Solution 59P Problem (a) To find the ratio of pressure amplitude Step 1: Frequency of Soprano f = 932 Hz 1 Density of air = 1.2 kg/m 3 5 Bulk modulus B = 1.42x10 Pa The intensity level is same Step 2: Pmaxb Ib = 2B Is P maxs 2B Where I s - Intensity of soprano Ib- Intensity of bass Here density of air and bulk modulus are constants. They cancel each other. The Intensity is same for both soprano and bass Step 3: Therefore 1 = P maxb P maxs P maxs = P maxb The ratio of Pressure amplitude of bass to the soprano = 1 Problem (b) To find the ratio of displacement amplitude of bass to that of soprano Step 1: Frequency of Soprano f s = 932 Hz Frequency of bass is 3 octave lower than the soprano 3 3 Frequency of Bass f b = f s2 = 932*2 Frequency of Bass f b = 116.5 Hz Step 2: B.A.2.f I = v v - velocity of sound B, v are constants The ratio of intensity is same Step 3: Cancelling constants and substituting Intensity value I A f I b = A fb s s s A b*16.5 1 = A *932 A b 932 A s = 116.5 A b = 8 A s ratio of displacement amplitude of bass to that of soprano = 8 Problem (c) Step 1: To find displacement amplitude Intensity level = 72dB Before finding displacement amplitude, we must find intensity and pressure amplitude. Step 2: Intensity I = I .10 /10 0 12 72/10 I = 10 .10 I = 1012.107.2 I = 1.58x10 5 W/m 2