In a large room, a person receives direct sound waves from a source $$120$$ metres away from him. He also receives waves from the same source which reach him, being reflected from the $$25$$ metre high ceiling at a point halfway between them. The two waves interfere constructively for wavelength of

Question

Infinity Learn · Accepted Answer

Let S be source of sound and P the person or listner.The waves from S reach point P directly following the path SMP and being reflected from the ceiling at point A following the path  SAP.  M is $$mid-point$$ of  SP (i$$.e. SM $$=$$ $$MP) and ∠ $$SMA=90o$$ Path difference between waves Δ$$x=SAP$$−SMP We have  $$SAP=SA+AP=2(SA)$$ $$=2$$[(SM)2+(MA)2$$]$$=2(602+252)=130m$$∴Path difference $$=$$ SAP – SMP $$=$$ $$130$$ $$-120$$ $$=$$ $$10$$ mPath difference due to reflection from ceiling $$=$$ λ$$2$$∴Effective path difference Δ$$x=10+$$λ$$2For$$ constructive interference Δ$$x=10+$$λ$$2=n$$λ⇒$$(2n$$−$$1)λ$$2=10(n=1$$,  $$2$$,  $$3$$....$$)$$∴Wavelength λ$$=2$$×$$10(2n$$−$$1)=202n$$−$$1$$. The possible wavelength are λ$$=20$$,  $$203$$,  $$205$$ ,  $$207$$ ,  $$209$$ ,  …..$$=$$ $$20$$ m, $$6.67$$ m, $$4$$ m, $$2.85$$ m, $$2.22$$ m, …..

In a large room, a person receives direct sound waves from a source 120 metres away from him. He also receives waves from the same source which reach him, being reflected from the 25 metre high ceiling at a point halfway between them. The two waves interfere constructively for wavelength of

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