Given data:

The differential pressure p(x,t) of a sound wave that is travelling in the x direction in air is 10 N/m2 at x=0 and t=50 μs.

The reference phase of p(x,t) is 36°.

The velocity of the sound in air is 330 m/s.

The frequency of the sound wave is 2 kHz.

Calculation:

The formula to calculate the angular frequency is given as,

ω=2πf

Here,

f is frequency of the sound wave and

ω is angular frequency.

Substitute 2×103 for f to obtain angular frequency.

ω=2π(2×103)=4π×103 rad/s

The formula for the wavelength of the sound wave is given as,

λ=upf

Here,

up is velocity of the sound in the air and

λ is the wavelength.

Substitute 2×103 for f and 330 for up in the equation to obtain the wavelength.

λ=3302×103=0.165 m

The formula for the phase constant is given as,

β=2πλ

Here, the phase constant is β.

Substitute 0.165 for λ to obtain the phase constant.

β=2π0.165=38.08

The expression for the differential pressure is given as follows.

p(x,t)=Acos(ωt−βx+ϕ0) (1)

Here,

p(x,t) is the differential pressure of a sound wave that is travelling in the x direction in air,

ϕ0 is the reference phase of p(x,t), and

t is time.

Substitute 10 N/m2 for p(x,t), 4π×103 rad/s for ω, 50×10−6 for t, 38.08 for β, 0 for x and 36° for ϕ0 in equation (1).

10=Acos(4π×103×50×10−6−38.0799×(0)+36°)10=Acos(0.6283−0+36°)10=Acos(0.6283×180°π+36°)10=Acos(36°+36°)

Further solving the above expression as,

10=Acos(72°)A=32.36

Substitute 32.36 for A, 4π×103 rad/s for ω, 38.08 for β, and 36° for ϕ0 in equation (1).

p(x,t)=32.36cos(4π×103t−38.08x+36°)

Conclusion:

Therefore, the complete expression for p(x,t) is,

p(x,t)=32.36cos(4π×103t−38.08x+36°).