Abstract
Our daily experience deals with sound waves, electromagnetic waves (as radio or light waves), deep or surface water waves, elastic waves in solid materials. Oscillatory phenomena manifest themselves also in contexts and ways less macroscopic and known. This is the case, for instance, of rarefaction and shock waves in traffic dynamics or of electrochemical waves in human nervous system and in the regulation of the heart beat. In quantum physics, everything can be described in terms of wave functions, at a sufficiently small scale.
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Notes
- 1.
See Segel [25].
- 2.
Consequence of absence of distributed moments along the string.
- 3.
It is the magnitude of a force.
- 4.
For instance, guitar and violin strings are nearly homogeneous, perfectly flexible and elastic.
- 5.
Recall that, at first order, if \( \epsilon \ll 1,\sqrt{1+\epsilon }-\simeq \epsilon \Big/2 \).
- 6.
Which obeys Hooke’s law: the strain is a linear function of the stress.
- 7.
For instance:
$$ F\left(x+ ct\right)=\frac{1}{2}g\left(x+ ct\right)+\frac{1}{2c}{\displaystyle {\int}_0^{x+ ct}h(y) dy} $$and
$$ G\left(x- ct\right)=\frac{1}{2}g\left(x- ct\right)+\frac{1}{2c}{\displaystyle {\int}_{x- ct}^0h(y) dy.} $$.
- 8.
In fact they are the characteristics for the two first order factors in the factorization (6.15).
- 9.
In the movie The Legend of 1900 there is a spectacular demo of this phenomenon.
- 10.
Check it.
- 11.
E.g. C 2 functions.
- 12.
Remember that
$$ a{x}^2+2 bxy+c{y}^2=a\left(x-{x}_1\right)\left(x-{x}_{{}^2}\right) $$where
$$ {x}_{1,2}=\left[-b\pm \sqrt{b^2- ac}\right]\Big/a. $$.
- 13.
It is understood that all the functions are evaluated at x = Φ (ξ, η) and t = Ψ(ξ, η).
- 14.
I.e. they can be locally expanded in Taylor series.
- 15.
Appendix D.
- 16.
Appendix D.
- 17.
Thanks to the miraculous presence of the factor 2 in the coefficient of w’!.
- 18.
As usual we can afford corner points (e.g. a triangle or a cone) and also some edges (e.g. a cube or a hemisphere).
- 19.
The tension T has the following meaning. Consider a small region on the membrane, delimited by a closed curve γ. The material on one side of γ exerts on the material on the other side a force per unit length T (pulling) along γ. A constitutive law for T is
$$ \mathbf{T}\left(x,y,t\right)=t\left(x,y,t\right)\mathbf{N}\left(x,y,t\right)\kern1.32em \left(x,y\right)\in \gamma $$where N is the outward unit normal vector to γ, tangent to the membrane. Again, the tangentiality of the tension force is due to the absence of distributed moments over the membrane.
- 20.
The two ratios must be equal to the same constant. The choice of −λ2 is by our former experience.
- 21.
We leave to the reader to find appropriate smoothness hypotheses on h, in order to assure that (6.60) is the unique solution.
- 22.
We do not expect incoming waves, which should be generated by sources placed far from the piston.
- 23.
Here δ is one dimensional.
- 24.
Check it, mimicking the proof in dimension one (Section 6.4.2).
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Salsa, S., Vegni, F.M.G., Zaretti, A., Zunino, P. (2013). Waves and vibrations. In: A Primer on PDEs. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2862-3_6
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