Sine waves

1. May 8, 2010

roam

1. The problem statement, all variables and given/known data

A conduction heat wave is caused to pass through a metal bar of average temperature Tmean = 35.0 °C, altering its temperature sinusoidally with an amplitude of Ti = 2.0 °C. The equation which gives the temperature, T(x,t), of the metal bar at any place x (in meters) inside it at any time t (in seconds) is:

T(x,t) = Tmean + Ti sin[ 2π(0.027t – 3.0x) + Co ]

where Co = 1.0π.

(a) What is the amplitude of the wave? (b) What is the wavelength of the wave?

2. Relevant equations

• Speed of sound wave (where B is the bulk modulus and mu p is the pressure): $$v=\sqrt{\frac{B}{\rho}}$$
• Power of sinusoidal wave: $$P=\frac{1}{2} \mu \omega^2 A^2 v$$
• For sound traveling through air:$$v=(331)\sqrt{1+\frac{T_C}{273}}$$

• Wave function for a sinusoidal wave: $$y=A sin(kx-\omega t)$$

3. The attempt at a solution

I don't understand how to apprach this problem. For example for part (a), what formula can I use? Also how do I find the temprature using the given equation:

T(x,t) = Tmean + Ti sin[ 2π(0.027t – 3.0x) + Co ]

what values do I need to substitute for "x" and "t"?

Last edited: May 8, 2010
2. May 9, 2010

diazona

Where does it ask you to find the temperature?
The temperature is different at different positions and different times. To find the temperature at a specific point in space and at a specific time, you would substitute in the position coordinate (x) of that point in space, and the time coordinate t.

Part (a) asks you for the amplitude of the wave. Do you know what amplitude means? Do you know how to find it from a wave equation? For instance, the equation
$$y=A \sin(kx - \omega t)$$
represents a wave. What would be the amplitude of that wave?

3. May 9, 2010

roam

"A" represents the amplitude. But how should I find the amplitude in this particular problem? The equation given only describes the temprature not the wave.

4. May 9, 2010

diazona

Well, you know that
$$y(x,t)=A \sin(kx - \omega t)$$
represents a wave, right? Do you also accept that
$$y(x,t) = y_\text{mean} + A \sin(kx - \omega t + \phi)$$
represents a wave? (How would you find its amplitude?)

If so, what's the problem? The equation
$$T(x,t) = T_\text{mean} + T_i \sin[2\pi(0.027t - 3.0x) + C_0]$$
is exactly the same thing, just with different letters.

5. May 10, 2010

roam

Thanks, I get it now.

Here's my last question: they further ask "What is the value of T(x,t) when t = 170.0 s, and x = 210.0 mm?"

I simply substituted the given values of "t" and "x" (along with other previously given values) into the equation:

T(x,t) = Tmean + Ti sin[ 2π(0.027t – 3.0x) + Co]

But the value I got was not the correct answer. Why is that?

6. May 10, 2010

Killeregg

did u convert into metres?

7. May 11, 2010

roam

40 mm = 0.04 m

$$37 sin(2 \pi (0.027(130)-3(0.04)+)\pi) = -23.5$$

But the correct answer must be 33.7! Why??