Question

In Fig. 17-46, sound of wavelength $\mathbf{0}\mathbf{.850}\mathbf{\text{\hspace{0.17em}m}}$is emitted isotropically by point source S. Sound ray 1 extends directly to detector D, at distance $\mathit{L}\mathbf{}\mathbf{=}\mathbf{10.0}\mathbf{\text{\hspace{0.17em}m}}$. Sound ray 2 extends to Dvia a reflection (effectively, a “bouncing”) of the sound at a flat surface. That reflection occurs on a perpendicular bisector to the SDline,at distance dfrom the line. Assume that the reflection shifts the sound wave by$\mathbf{0}\mathbf{.500}\mathbf{\lambda }$. For what least value of d(other than zero) do the direct sound and the reflected sound arrive at D(a) exactly out of phase and (b) exactly in phase?

Short answer

1. The least value of d for which the direct and reflected sounds arrive at D exactly out of phase is,$2.10\text{\hspace{0.17em}m}$.
2. The least value of d for which the direct and reflected sounds arrive at D exactly in phase is $1.47\text{\hspace{0.17em}m}$.

Step-by-step solution

The given data

1. Wavelength of sound emitted isotopically by point source is$0.850\text{\hspace{0.17em}m}$.
2. Sound ray 1 distance to detector D,$\mathrm{L}=10.0\text{\hspace{0.17em}m}$.
3. Reflection shifts the sound wave by$0.500\mathrm{\lambda }$

Understanding the concept of interference 

We can find the path difference between the direct and reflected waves. Then using the conditions for constructive and destructive interference, we can find the least value of d, for which the direct and reflected sounds arrive at D exactly in phase and out of phase.

Formulae:

The cosine law for side c of triangle,

${\mathbf{c}}^{\mathbf{2}}\mathbf{=}{\mathbf{a}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathbf{abcosC}$ …(i)

The linear expansion formula,

$\mathbf{L}\mathbf{=}{\mathbf{L}}_{\mathbf{0}}\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{\alpha \Delta T}\mathbf{)}$ …(ii)

a) Calculation of least value of d for destructive interference 

Path difference between direct and reflected wave using equations (i) and (ii) is given as:

$\begin{array}{c}\mathrm{\Delta x}=\sqrt{{\mathrm{L}}^2+{\left(2\mathrm{d}\right)}^2}-\mathrm{L}+0.500\mathrm{\lambda }\\ =\sqrt{{\left(10\text{\hspace{0.17em}m}\right)}^2+{\left(2\mathrm{d}\right)}^2}-10\text{\hspace{0.17em}m}+0.500\left(0.850\text{\hspace{0.17em}m}\right)\\ =\sqrt{{\left(10\text{\hspace{0.17em}m}\right)}^2+{\left(2\mathrm{d}\right)}^2}-9.575\text{\hspace{0.17em}m}\end{array}$

For destructive interference, the least value of d is given as:

$\begin{array}{c}\frac{\mathrm{\Delta x}}{\mathrm{\lambda }}=0.5,1.5,\dots .\\ \frac{\sqrt{{\left(10\text{\hspace{0.17em}m}\right)}^2+{\left(2\mathrm{d}\right)}^2}-9.575\text{\hspace{0.17em}m}}{0.850\text{\hspace{0.17em}m}}=0.5,1.5,\dots .\\ \sqrt{{\left(10\text{\hspace{0.17em}m}\right)}^2+{\left(2\mathrm{d}\right)}^2}-9.575\text{\hspace{0.17em}m}=0.425\text{\hspace{0.17em}m},1.275\text{\hspace{0.17em}m},\dots .\\ \sqrt{{\left(10\text{\hspace{0.17em}m}\right)}^2+{\left(2\mathrm{d}\right)}^2}=10\text{\hspace{0.17em}m},10.85\text{\hspace{0.17em}m},\mathrm{..}\\ {\left(10\text{\hspace{0.17em}m}\right)}^2+{\left(2\mathrm{d}\right)}^2=100\text{\hspace{0.17em}m},117.72\text{\hspace{0.17em}m},\mathrm{...}\\ \mathrm{d}=0,2.1\text{\hspace{0.17em}m}\mathrm{...}\end{array}$

Hence the value of d is,$0,2.1\text{\hspace{0.17em}m}\mathrm{...}$.

Excluding zero, the least value is found to be$\mathrm{d}=2.10\text{\hspace{0.17em}m}$.

Therefore, the least value of d for which the direct and reflected sounds arrive at D exactly out of phase is $2.10\text{\hspace{0.17em}m}$.

b) Calculation of least value of d for constructive interference

For constructive interference, the least value of d is given as:

$\begin{array}{c}\frac{\mathrm{\Delta x}}{\mathrm{\lambda }}=1,2,\dots .\\ \frac{\sqrt{{\left(10\text{\hspace{0.17em}m}\right)}^2+{\left(2\mathrm{d}\right)}^2}-9.575\text{\hspace{0.17em}m}}{0.850\text{\hspace{0.17em}m}}=1,2,\dots .\\ \sqrt{{\left(10\text{\hspace{0.17em}m}\right)}^2+{\left(2\mathrm{d}\right)}^2}-9.575\text{\hspace{0.17em}m}=0.850\text{\hspace{0.17em}m},1.7\text{\hspace{0.17em}m},\dots .\\ \sqrt{{\left(10\text{\hspace{0.17em}m}\right)}^2+{\left(2\mathrm{d}\right)}^2}=10.425\text{\hspace{0.17em}m},11.275\text{\hspace{0.17em}m},\mathrm{...}\\ {\left(10\text{\hspace{0.17em}m}\right)}^2+{\left(2\mathrm{d}\right)}^2=108.68\text{\hspace{0.17em}m},127.126\text{\hspace{0.17em}m},\mathrm{..}\\ \mathrm{d}=1.47\text{\hspace{0.17em}m},2.6\text{\hspace{0.17em}m,}\mathrm{...}\end{array}$

Solving this, we get the least value of d as:

$\mathrm{d}=1.47\text{\hspace{0.17em}m}$

Therefore, the least value of d for which the direct and reflected sounds arrive at D exactly in phase is $1.47\text{\hspace{0.17em}m}$.