Overtones in a string (equation for wave)

[Incand]

Homework Statement

A string has the fundamental tone of
$s_1 = A_1\sin (\omega_1 t - k_1 x)$
Determinate the wave equation for the first harmonic of the string if the sound intensity level of the harmonic is 20dB lower than the fundamental tone. $\omega_1 = 1360/s$ and $k_1 = 4/m$.

Homework Equations

Sound intensity level
$L = 10\lg (\frac{I_1}{I_0} )$
Sound intensity is proportional to the wave amplitude squared

$I \sim A^2$

The Attempt at a Solution

Sound intensity level
$-20 = 10\lg( \frac{I_2}{I_1}) \Longleftrightarrow I_2 = 0.01I_1$
Amplitude relationship
$\frac{A_2}{A_1} = 0.1 \Longleftrightarrow A_2 = 0.1A_1$
The first harmonic got double the frequency so
$s_2 = 0.1A_1 \sin(2 \omega_1 t - 2 k_1 x)$
which is wrong. According to the answer key it should be
$s_2 = 0.05A_1\sin(2 \omega_1 t-2k_1 x)$
Why is it 0.05?

 

[TSny]

Energy of a wave on string depends on $\omega$ as well as $A$.

See http://hyperphysics.phy-astr.gsu.edu/hbase/waves/powstr.html

 

[CWatters]

As I recall the intensity of a wave is proportional to the square of its amplitude. But perhaps I'm rusty.

 

[Incand]

Energy of a wave on string depends on $\omega$ as well as $A$.

See http://hyperphysics.phy-astr.gsu.edu/hbase/waves/powstr.html

Cheers!

As I recall the intensity of a wave is proportional to the square of its amplitude. But perhaps I'm rusty.

Always a bit tricky, luckily we got TSny to link us the formula :)