**Problem 1**. Calculate the fundamental frequenet; of a quartz crystal of 2 mm thickness. The density of the crystal is 2650 kg/m3 and Young’s modulus is 7.9 x 1010 N/m2.

**Hint:** [Ans. 1.365 MHz]

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**Problem 2**. An ultrasonic beam of 1 em wavelength sent by the ship returns from sea bed after 2 seconds. If velocity of ultrasonic beam in sea water is 1510 m/sec at 0°C, its salinity at 30°C is

2 gm/lit., calculate the depth of sea bed at 30°C.

**Hint**: v1 = v0 + 1.14s + 4.21t- 0.037 t^{2}

and Depth of sea bed at 30°C = V_{30}*t/2

[Ans. 1637.20 m]

**Problem 3.** The difference between the transmission time and receiving time is 0.44 s. Calculate the depth of sea bed in this case if velocity of ultrasonic wave is 1440 m/sec.

**Hint: **2d/ t = v

[Ans.d=316.8m]

**Problem 4.** A quartz crystal of thickness 0.001 m is vibrating at resonance. Calculate the fundamental frequency. Given Y for quartz 7.9 x 1010 N/m2 and p for quartz = 2650 kg/m3.

**Hint: **

[Ans. n = 2.73 MHz]

**Problem 5**. A piezoelectric X-cut quartz plate has a thickness of 1.5 mm. If the velocity of propagation of longitudinal sound waves along the X-direction is 5760 m/s, calculate the fundamental frequency of the crystal.

Hint: Thickness = λ/2 , n = v/λ