by Leslie Greengard and Tonatiuh Sánchez-Vizuet
Cathleen Morawetz was a force at the Courant Institute when one of us (L.G.) arrived as a postdoctoral fellow. It was the last year of her directorship, but she made the time to welcome all newcomers. Her generosity of spirit was unmatched — she encouraged young people in every discipline, and her humour and enthusiasm were infectious.
When she began to study the decay properties of acoustic waves after
impinging on an obstacle, essentially no general results were
available. To understand the relevant issues, let us begin with the
formulation of the problem in terms of the governing linear, scalar
wave equation in

Suppose now that, rather than propagating in free-space, the outgoing
spherical wavefront emanating from
Let

In 1961, Morawetz
[1]
made a critical step forward. She showed that if
the reflecting obstacle is star-shaped, then the solution to the wave
equation decays like
The qualitative difference in the behavior of waves reflecting from obstacles that are not star-shaped and those that are is illustrated in Figures 3 and 4. In the first three panels of each figure, as the incoming wave hits the object, the scattered wave is clearly visible, with energy propagating outwards in all directions. In the next three panels, more of the energy is carried away. In Figure 3, some of the energy remains behind for quite some time, and in the last panel a significant amount of energy has focused in a small neighborhood. In Figure 4, the energy has propagated outward without significant concentration and appears to decay much more rapidly.

Morawetz’s writing style was very much that of a storyteller. To get a sense of that, here is the beginning of the proof of the main theorem in her 1961 paper [1]:
The proof is based on energy identities, i.e., quadratic integral relations satisfied by all solutions. This is one of the most powerful tools for getting estimates for solutions of elliptic, hyperbolic or mixed equations. The most familiar identity of this kind for the wave equation is obtained by multiplying
by and integrating in the slab ; the resulting integral identity satisfies the conservation of energy. Here we use another multiplier in the place of introduced by Protter for another purpose. The significance of using alternative multipliers has been frequently emphasized by Friedrichs and is often referred to as Friedrichs’s , , -method. The multiplier here is and from the resulting identity we conclude that all the energy is carried outward.
In truth, Morawetz was being overly modest. It was her keen insight that allowed for the selection of a multiplier which would yield the desired result. The power and generality of this approach led to breakthroughs in many wave propagation problems, with the state of the art collected in Morawetz’s 1966 monograph “Energy identities for the wave equation,” originally released as a Courant Institute technical report.
A second major step forward in understanding the
decay of waves scattered from star-shaped obstacles
came in 1963, in joint work with
Lax
and
Phillips.
They showed that, in fact, such solutions decay exponentially
(as they do for a sphere), not just as
Nontrapping objects

This situation was studied by Morawetz,
Ralston,
and
Strauss
in their
1977 article, where they proved a remarkable extension of Morawetz’s
earlier results; if the object
Denoting by
Geometric optics and frequency domain analysis
In the study of linear wave propagation, much of our
understanding comes from the frequency domain — that
is, analyzing the Fourier transform of the wave equation

Without entering into technical
details, geometrical optics is based on expanding the incoming and
scattered waves in terms of a series in inverse powers of the
wavenumber
Although Morawetz herself did little numerical computation, her analytic work (especially on multipliers) has played a major role in the design of numerical methods. We cannot do justice to the literature here, but refer the reader to three recent papers: one on eigenvalue computation, one on frequency domain scattering, and one on time-domain integral equations [e1] [e2] [e5]. We have only been able to scratch the surface of her legacy in this note. Her contributions are profound and deep, and have changed the way we think about partial differential equations. She was a wonderful friend and colleague and is greatly missed.