Bollettino di Geofisica Teorica e Applicata
OGS Website
About the Journal
Contacts
To Authors
On-line Submission
Subscriptions
Forthcoming
On-line First
The Historical First Issue
Issues

2022 Vol. 63
1 / 2

2021 Vol. 62
1 / 2 / 3 / 4 / Suppl. 1 / Suppl. 2 / Suppl. 3

2020 Vol. 61
1 / 2 / 3 / 4 / Suppl. 1

2019 Vol. 60
1 / 2 / 3 / 4 / Suppl. 1 / Suppl. 2 / Suppl. 3

2018 Vol. 59
1 / 2 / 3 / 4 / Suppl. 1

2017 Vol. 58
1 / 2 / 3 / 4

2016 Vol. 57
1 / 2 / 3 / 4 / Suppl. 1

2015 Vol. 56
1 / 2 / 3 / 4

2014 Vol. 55
1 / 2 / 3 / 4

2013 Vol. 54
1 / 2 / 3 / 4 / Suppl. 1 / Suppl. 2

2012 Vol. 53
1 / 2 / 3 / 4

2011 Vol. 52
1 / 2 / 3 / 4 / Suppl. 1

2010 Vol. 51
1 / 2-3 / 4 / Suppl. 1

2009 Vol. 50
1 / 2 / 3 / 4

2008 Vol. 49
1 / 2 / 3-4 / Suppl. 1

2007 Vol. 48
1 / 2 / 3 / 4

2006 Vol. 47
1-2 / 3 / 4

2005 Vol. 46
1 / 2-3 / 4

2004 Vol. 45
1-2 / 3 / 4 / Suppl. 1 / Suppl. 2

2003 Vol. 44
1 / 2 / 3-4

2002 Vol. 43
1-2 / 3-4

2001 Vol. 42
1-2 / 3-4

2000 Vol. 41
1 / 2 / 3-4

1999 Vol. 40
1 / 2 / 3-4

1998 Vol. 39
1 / 2 / 3 / 4

1997 Vol. 38
1-2 / 3-4

1995 Vol. 37
145 / 146 / 147 / 148 / Suppl. 1

1994 Vol. 36
141-144 / Suppl. 1

1993 Vol. 35
137-138 / 139 / 140

1992 Vol. 34
133 / 134-135 / 136

1991 Vol. 33
129 / 130-131 / 132

 
 

Vol. 63, n.2, June 2022
pp. 175-188

The Kramers-Kronig relations and the analogy between electromagnetic and mechanical waves

M. Carcione, F. Mainardi, J. Ba and J. Chen

Received: 28 October 2021; accepted: 21 November 2021; published online: 8 March 2022

Abstract

The important consequence of the Kronig-Kramers relations (KKrs) is that dissipative behaviour in material media inevitably implies the existence of dispersion, i.e. a frequency dependence in the constitutive equations. Basically, the relations are the frequency-domain expression of causality and correspond mathematically to pairs of Hilbert transforms. The relations have many forms and can be obtained with diverse mathematical tools. Here, two different demonstrations are given in the electromagnetic case, illustrating the eclectic mathematical apparatus available for this purpose. Then, we apply the acoustic (mechanical)-electromagnetic analogy to obtain the elastic versions. One major consequence is wave propagation attenuation and pulse spreading, that is, the progressive widening of a pulse as it propagates through a medium [vacuum seems to be the only "medium" where this does not occur (electromagnetic dispersion), while mechanical waves do not propagate]. Therefore, we derive KKrs that relate the wave velocity to the attenuation and quality factors. Finally, we discuss the concepts of stability and passivity and provide a novel algorithm to compute the relations numerically by using the fast Fourier transform.



Download PDF complete


back to table of contents