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Q79P

Expert-verifiedFound in: Page 511

Book edition
10th Edition

Author(s)
David Halliday

Pages
1328 pages

ISBN
9781118230718

**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 ${\mathit{L}}{\mathbf{}}{\mathbf{=}}{\mathbf{10.0}}{\mathbf{\text{\hspace{0.17em}m}}}$**

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

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

**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)**

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 $2.10\text{\hspace{0.17em}m}$.

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 $1.47\text{\hspace{0.17em}m}$.

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