Interference wave have solution need explanation

[Puchinita5]

Homework Statement

In Fig. 17-35, sound with a 46.4 cm wavelength travels rightward from a source and through a tube that consists of a straight portion and a half-circle. Part of the sound wave travels through the half-circle and then rejoins the rest of the wave, which goes directly through the straight portion. This rejoining results in interference. What is the smallest radius r (cm) that results in an intensity minimum at the detector? (Take the speed of sound to be 343 m/s.)

Homework Equations

http://edugen.wiley.com/edugen/courses/crs1650/art/qb/qu/c17/pict_17_34.gif

The Attempt at a Solution

i have the solution...what i don't understand is WHY the path length difference equals 23.2? don't see how or why it's half the wavelength?

 

[Hootenanny]

HINT: A minimum means that destructive interference has occurred ...

 

[tomcue]

Interference is a phenomenon that occurs when two or more waves interact with each other. In this case, the sound wave is traveling through a tube and is encountering a half-circle obstacle. When the wave reaches the half-circle, it is split into two parts - one part travels through the half-circle and the other continues through the straight portion of the tube. When these two parts of the wave rejoin, they interfere with each other, resulting in an intensity minimum at the detector.

The path length difference is the difference in distance that the two parts of the wave have traveled before rejoining. In order for destructive interference to occur, the path length difference must be equal to half of the wavelength. This is because when the two waves rejoin, they are out of phase by half a wavelength, resulting in cancellation of the wave.

In this problem, the sound wave has a wavelength of 46.4 cm. Therefore, in order for the path length difference to be half of the wavelength, the difference must be 23.2 cm. This means that the radius of the half-circle must be such that the distance traveled by the wave through the half-circle is 23.2 cm longer than the distance traveled through the straight portion of the tube. This can be calculated using the speed of sound and the equation for circumference of a circle.

I hope this explanation helps clarify the concept of interference and the reasoning behind the path length difference in this problem. If you have any further questions, please don't hesitate to ask.